CONTINUUM STRUCTURAL REPRESENTATION OF FLEXURE AND TENSION STIFFENED ONE-DIMENSIONAL SPACECRAFT ARCHITECTURES. Jeffrey James Larsen

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1 CONTINUUM STRUCTURAL REPRESENTATION OF FLEXURE AND TENSION STIFFENED ONE-DIMENSIONAL SPACECRAFT ARCHITECTURES by Jeffrey James Larsen A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering MONTANA STATE UNIVERSITY Bozeman, Montana May, 2009

2 c Copyright by Jeffrey James Larsen 2009 All Rights Reserved

3 ii APPROVAL of a thesis submitted by Jeffrey James Larsen This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the Division of Graduate Education. Dr. Christopher H. M. Jenkins Approved for the Department of Mechanical Engineering Dr. Christopher H. M. Jenkins Approved for the Division of Graduate Education Dr. Carl A. Fox

4 iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfullment of the requirements for a master s degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. If I have indicated my intention to copyright this thesis by including a copyright notice page, copying is allowable only for scholarly purposes, consistent with fair use as prescribed in the U.S. Copyright Law. Requests for permission for extended quotation from or reproduction of this thesis in whole or in parts may be granted only by the copyright holder. Jeffrey James Larsen May, 2009

5 iv DEDICATION I dedicate this work to my parents. They have shown me the true value of hard work and determination. I am forever indebted to them.

6 v ACKNOWLEDGEMENTS I would like to thank Dr. Christopher Jenkins for introducing me to research as an undergraduate and opening my eyes to a level of schooling I had never considered before my senior year. I am deeply thankful for all of his help, insight, and time that he put into this project. I would also like to thank Jeremy Banik for his help and input on the project and allowing me to work with him for a great summer in Albuquerque. Further thanks is necessary to thank those who provided time and energy helping with this project; Dr. Doug Cairns and Dr. Ladean McKittrick of Montana State University, and Dr. Thomas Murphey of the Air Force Research Lab. Funding Acknowledgment The work herein was supported in part by the Air Force Research Laboratory Space Vehicles Directorate through contract and the Summer Space Scholar program.

7 vi TABLE OF CONTENTS 1. INTRODUCTION...1 Structural Architectures...1 Dimensional Architectures...2 One-dimensional Representation...5 Design Impact LITERATURE REVIEW...7 Vibrations...7 String-Beam Systems Deployable Structures FINITE ELEMENT MODEL ABAQUS Support Model Setup ABAQUS Support-Payload Model Setup Payload Support Model for N greater than MATHEMATICAL MODEL Model Development Coupled Beam-Sting Equations Support Equation Boundary Conditions Support Solution Payload Equation Payload Boundary Conditions Payload Solution System Coupling System Solution Decoupled Boundary Conditions Beam-Beam Equation Derivation Beam-Beam Support Derivation Beam-Beam Payload Derivation RESULTS Numerical Results Analytical Results Numerical vs Analytical Expanded Numerical Results... 43

8 vii TABLE OF CONTENTS CONTINUED System Relationships CONCLUSION REFERENCES CITED APPENDICES APPENDIX A: ABAQUS FEM Code APPENDIX B: Solution and Constant Validation... 68

9 viii Table LIST OF TABLES Page 1 Verification of ABAQUS Beam Model Member specific material properties and general system properties... 40

10 ix Figure LIST OF FIGURES Page 1 Examples of stiffened architectures with (a) tension and (b) flexure Examples of structural architectures for (a) one-dimension and (b) twodimensions General representation of one-dimensional architecture model General configuration to encompass all possible 1 and 2 dimensional architectures General representation of the payload-support interaction Sytem variable relationships and interactions Beam-String model developed for nonlinear analysis Beam-String model developed for a fiber optic coupler Model Representation of ABAQUS Support only model Comparison and validation of ABAQUS approach General Representation of Support-Payload model for N= ABAQUS representation of the interaction between the payload and support for N= The general support-payload model for N = The representation of the ABAQUS support-payload model for N = Forces and moments acting on a differential element of the beam Comparison of system frequency based on a full range of load and mass ratios Comparison of system frequency for mass ratios of interest Sample curve of the characteristic equation to determine zero crossings for the beam-string model Frequency values of the beam-string model over range of load ratios Sample curve of the characteristic equation to determine zero crossings for the beam-beam model Frequency values of the beam-beam model over range of load ratios... 42

11 x Figure LIST OF FIGURES CONTINUED Page 22 Results for Mass Ratio of 100: Results for Mass Ratio of 10: Results for Mass Ratio of 1: Results for Mass Ratio of 1: Changes in system mode shape as the payload tension is increased for a mass ratio of 1: Comparison of individual member frequencies to system frequency Effects of changes in system variables Variation of length as the total mass is increased at a fixed frequency (f 1 > f 2 > f 3 ) Comparison of numerical data with approximate equation Variation of frequency as the number of ties is increased Converging frequency curves for several mass ratios as the number of ties is increased Effect of number of ties on frequency for a range of load ratios... 55

12 xi NOMENCLATURE A Cross-section Area, m 2 a Payload Load Parameter E Elastic Modulus, Pa f Frequency, Hz i Mode Number I Area Moment of Inertia, m 4 k Load Parameter L Length, m M Total Mass, kg MR Mass Ratio also written as m s : m p m Component Mass, kg N Number of Ties t Time,sec P Axial Load, N P cr Critical Buckling Support Load, N w Displacement, m β Non-Dimensional Frequency Value λ Frequency Parameter µ Mass per unit length, kg/m ω Frequency, rad/s ρ Density, kg/m 3 Subscript p Payload Component s Support Structure Component

13 xii ABSTRACT Spacecraft designs are a result of system properties and design variables that ensure the spacecraft will operate to mission objectives. The focus of this effort is a set of global system variables for frequency, length, total mass and the ratio between the payload mass and the support structure mass. These properties will be explored to observe the behavior of the system and develop relationships that govern the trade-offs between the variables and assist mission planners in future spacecraft design. These variables will be observed in one-dimensional structures where the dominating dimension is many times larger than the other two dimensions and the system is comprised of a support and a payload member. To observe the interaction between the payload and the support, the system was varied for different system variables and observed through ABAQUS finite element software. Attempts were made to predict the system frequency through mathematical approaches. The finite element work was able to generate several approximate relationships between the system variables and the fundamental natural frequency of the system. From these relationships an approximate equation was developed for the frequency for a fixed mass ratio and load ratio as a function of the length, bending stiffness, and total mass of the system. Additional work into the changes to the system as the number of connect points is increased shows the system converging towards a frequency solution which results in a minimized dependence on the connection points. These results were then compared to those of several derived analytical models to determine if a closed-form solution could be used to predict system behavior over the same range of structural characteristics. This closed form solution proved to correlate well to analytical predictions only for the case where the support structure dominates the total system mass, and thus the structural system performs like a beam under compression. Further work is necessary to accurately predict the system frequency through an analytical approach. These insights promise to aid mission designers in objectively evaluating new structural architectures based on structural performance rather than on an unbalanced adherence to heritage or in some cases personal preference.

14 1 INTRODUCTION Structural Architectures In spacecraft design and modeling, several variables are key influences to selecting the proper structural architecture for a specific application. Most spacecraft have deployable appendages, blankets, or panels included to serve mission objectives. These deployable components can be designed in a variety of geometries and configurations. Examples of these include solar sails, sun shades, antennas, solar arrays, and phased arrays. These deployable payloads are often supported through two main stiffening methods, tension and flexure. These methods are often used independently but can be used in combination depending on the design of the structure. This selection of the stiffening method is defined as the structural architecture. The requirements of the system often dictate how the design will support the payload. Figure 1 shows two examples of how these stiffening methods are used in practice. In Figure 1a the Space Station Solar Arrays uses the tensioned in the arrays to create a compression in the mast of the system which effectively stiffens and supports the arrays. Figure 1b shows the Radar Sat II which carries solar arrays through a backing structure that uses the high bending stiffness to support the array. A combination of the two methods would incorporate both the generated compressive force and the present bending stiffness to reach the necessary support for the payload. The selection of the structural architecture can be further seen by observing the interaction between the payload and the support. Figure 1a can be seen to be connected in two places between the payload (the solar array) and the support (the mast) at both ends. The solar array has no bending stiffness but is tensioned to allow for the solar panels to effectively collect the sunlight. Similar to the bow and cord concept [3],

15 2 (a) Tension stiffening of the Space Station Solar Arrays[1] (b) Flexure stiffening of the RadarSat II[2] Figure 1: Examples of stiffened architectures with (a) tension and (b) flexure as the cord is pulled back the stiffness of the bow is increased. This tension causes an equal and opposite compression in the mast which creates the required stiffness. The stiffness generated in the RadarSat II occurs from the backing structure that holds the solar array. Picture the back side of Fig. 1b as the bow and cord example but with an infinite number of connections. Now the payload (the cord) becomes inherently stiffness from the bending stiffness of the bow. By changing the properties of the support the stiffness of the system can be modified. The potential for a combined stiffness method can be had by varying the number of connection points between these two and infinite boundaries. Dimensional Architectures The architectures can be further broken down into the dimensional components the system occupies. Of interest here are the one and two dimensional architectures. The one-dimensional space structure is defined where the linear dimension is many times greater than the width or thickness. DARPA s Innovative Space-based radar Antenna

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a)0+97% 4*4G.+)*0:% 90% DARPA s ISAT[4] (b) DLR Solar Sail[5] I98O%>X%'-7+.%'+97%FB773%5*17-3*5% D/8% U?:!:>VO% 6$*% 4+9)% 0#.B2#B.+7% *7*4*)#0% -F% #$*% P+11.-TO%!C4%T%!C4Q% E*17-34*)#% =-5B7*% Figure +.*% 4+5*% -F% <IJK%of 1+.#0% #-% architectures for (a) one-dimension and (b) two2: Examples structural &+2$%0+97%0*84*)#%90%+%.98$#%90-02*7*0%#.9+)87*%1.-R959)8% 49)949W*%4+00O% dimensions +)% +.*+% -F% "!OS% 4\O% 6$*% 09W9)8% -F% #$*% 0*84*)#0% L+0% G+0*5%-)%#$*%#.9+)8B7+.%01+2*%1.-R95*5%G*#L**)%*+2$%-F% Technology (ISAT) [4], Fig. 2a, is an example of <IJK% a one-dimensional The-F%two#L-%?M4% G--40% L9#$% structure. 2-)095*.+#9-)% 27*+.+)2*% +)5% 2-))*2#9-)%.*NB9.*4*)#0O% 6$.**% 59FF*.*)#% F974% dimensional space structure is similarly defined where the linear and width dimensions 4+#*.9+70%L*.*%B0*5%F-.%#$*%0+97%0*84*)#0X% are many times greater than the thickness and the linear and width dimensions are!!?!oc%"4%=37+.:%(7@2-+#*5%-)%-)*%095*% of a similar order of magnitude. The!! DLR]OD%"4%^+1#-):%(7@2-+#*5%-)%G-#$%095*0% solar sail [5], Fig 2b, is a good example of!! MOC%"4%K&_%PK-73*#$37*)@)+1$#$+7+#*Q:%% (7@2-+#*5%-)%G-#$%095*0O% the two-dimensional structure. 6$*%4+9)%-G`*2#9R*%F-.%B09)8%59FF*.*)#%4+#*.9+70% As a result of the stiffening methods, and the variation of the dimensions to al+0%#-% +00*00%#$*%F974%$+)579)8%+)5%1.-2*009)8:%+)5%#-%*R+7B+#*% 4+#*.9+7% G*$+R9-.% F-.% 0*+49)8:% F-759)8:% +0% L*77% +0% degree, a general architecture can be#$*% created to encompass a vast majority of these F-.%0+97%5*17-34*)#O%! space structures. Then by simply modifying the properties of this general architecture I98O%?X%'-7+.%'+97%E*17-34*)#%=-5B7*% (0% +% *% G*#L**)% 0*2B.*5% #.+)01-.#+#9-)% 9)#-% 01+2*:% Figure 2-)#.-77*5% 5*17-34*)#% +)5% #$*% 1*.F-.4+)2*% 9)% a specific design can be effectively modeled. 3a shows the one-dimensional case 6$*% 5*17-34*)#% L+0% 2+..9*5% -B#% B)5*.% 094B7+#*5% C@8% 5*17-3*5% 2-)F98B.+#9-)% 5*17-3+G7*% G--40% PB)2-97*5% )5% +4G9*)#% *)R9.-)4*)#+7% L9#$9)% 2+O% where 2-)59#9-)0% the bars between the >C% payload and represent the.*9)f-.2*5% connection F.-4%support +%.-77Q:%members 4+5*% -F% 2+.G-)% F9G.*% 17+0#920% 49)B#*0% P?D49)% F-.% Y--4% 5*17-34*)#:%?D49)% F-.% '+97% P<IJKQ:% L*.*% 2$-0*)% F-.% #$*% G+0*79)*% &'(;EZJ% '-7+.% *17-34*)#Q% +#% #$*% &B.-1*+)% <*)#*.% +#% EZJ% points.(0#.-)+b#% This system can quickly and easily be modified for the tension or flexural '+97% 5*098)O% 6$*% G--40% 2-)090#% -F% #L-% 7+49)+#*5% <-7-8)*O%I98%!%0$-L0%#$*%0#+#B0%L$*.*%#$*%G--40%L*.*% F7*T9G7*%#@0$+1*5%0$**#0%L$92$%+.*%G-)5*5%+#%#$*%*58*0% stiffening mechanisms by.9895% changing the number of ties. The Space Station Solar B773% 5*17-3*5% #-% +% 7*)8#$% -F%?M4:% 1.-R959)8% #$*% #-% F-.4% +% #BGB7+.% 0$+1*% P0**% I98O% MQ% UM:D:SVO% 6$*3% #.B2#B.*%F-.%#$*%0BG0*NB*)#%0+97%4*4G.+)*%5*17-34*)#O% Arrays, shown in Fig. 1a, can be treated as a one-dimensional tensioned 2-4G9)*% 0#.*)8#$% +)5% 0#9FF)*00% L9#$%architecture 7-L% 5*)09#3% +)5:% 1.*00*5%F7+#%+)5%2-97*5%+.-B)5%+%2*)#.+7%$BG%F-.%0#-.+8*% and would be represented by Fig. 3b L9#$9)% with two connection points. +% #98$#% R-7B4*:% #$*3% 2+)% G*% B)2-97*5% F.-4% #$*% 2*)#.+7%$BG%F-.%5*17-34*)#O%% The two-dimensional representation shown in Fig. 4a shows a top down view of the system with the payload and connection points to the support shown. The DLR solar

17 4 +,-.$,/ '()*(%!"##$%& '()*(% (a) One-dimension general case (b) One-dimension specific case Figure 3: General representation of one-dimensional architecture model (a) Two-dimension general case (b) Two-dimension specific case Figure 4: General configuration to encompass all possible 1 and 2 dimensional architectures sail, Fig 2b, could be mapped by reducing the general case to Fig. 4b. The general case can be modified to accommodate a wide range of shapes including circles, rectangles, and triangles. By gaining insight into these general cases system designs could easily work with the different architectures to observe the best design out of an array of varying options.

18 5 "677.$#(12342$ *+,-.+/(12342$ Figure 5: General representation of the payload-support interaction One-dimensional Representation The focus of this work investigates the tradeoffs between critical design variables for the one-dimensional structural architectures. The system is treated as acting along a single axis to simplify the model as shown in Fig. 5. This allows a tensioned payload to create a compressive force in the support member without concern for significant deflections occurring in either member. To aid in the ease of understanding the same model is shown in Fig. 6 where the payload and support are separated. This allows for the the individual payload and support components to be shown and how each contributes to the overall global variables. Each component contributes a bending stiffness, length, and mass to the system. The bending stiffness, represented by EI, where E is the elastic modulus and I is the area moment of inertia, will always be present for the support structure but can be set to zero for payloads that provide no bending stiffness, such as tensioned membranes. The payload and support components are tied together with N number of ties depending on system needs. In addition to the stiffness achieved through the bending stiffness, the system can also be stiffened though a tensioned payload, the tension defined by P. Based on these variables, the global system can be defined through length, L, total mass, M, the mass ratio between the support mass m s and the payload mass m p, m s :m p, and the overall system frequency f.

19 6!"#$%"&' -./0 * 1213 * '!' 7'!' 21'51' *!"#'!' ()**%+,' -./ '!' Figure 6: System variable relationships and interactions, shown for N = 5. Design Impact By modifying the component variables, number of ties, and tension force the changes to the global system performance can be observed. The focus of the system performance is on the system natural frequency as this has the biggest impact on the operating conditions of the spacecraft and is less predictable than the other system variables. Length, total mass, and mass ratio are often dictated by the mission requirements but potentially have some flexibility for a given system. These values will be modified to observe the changes to the frequency and look for trends between these changes. Further work will be developed through numerical and analytical approaches to gain insight and attempt to predict these outcomes. This work is the start of a new set of design tools for spacecraft designers. By giving flexibility to the designers the work will allow for a variety of options to be investigated and considered before the design selection occurs. It aims to break down some of the design approaches that are based on heritage rather than new options. This work focuses on the one-dimensional architecture to gain an understanding of how these tools can be developed through the simplest case. With this insight the work will be expanded into the two-dimensional architecture which will be able to incorporate the majority of existing designs.

20 7 LITERATURE REVIEW Vibrations A focus of this thesis has been on the fundamental operating frequency of the payload and support members of the designs. Previous work can be seen in several areas of published literature. Derivations and work with individual beams, work on coupled systems between beams and strings, and work in one-dimensional architectures. Initially, the work started in independent fashion using closed form frequency models and then coupled as a system using numerical analysis and minor references to previous work. This work could then be applied to existing models and designs. Numerous works have investigated the vibrations of beams and the effect of axial loads on the frequency. Gorman [6] presented a concise summary of vibrational analysis of beams with various boundary conditions and problem variations. Gorman presents detailed derivations and tabular data associated with each condition. Building on Gorman s work, Belvins [7] presents formulas for the frequency of a variety of structures and fluids, of interest here is the sections on cables and beams. For straight cables, similar to treating the payload with bending stiffness, Belvins notes the frequency as Eq. 1: f i = i ( ) 1/2 P i = 1, 2, 3,..., (1) 2L µ where µ is the mass per unit length of the beam and the index i is the mode number. The focus of the cables section in Belvins however is on the influence of sag in the cable on the frequency. The thorough section on beams provides several useful formulas. For a simple single span beam with free-free boundary conditions the natural

21 8 frequency is given as Eq. 2: f i = λ2 i 2πL 2 ( ) 1/2 EI i = 1, 2, 3,..., (2) µ where the parameter λ i is numerically determined. Of interest for this project is the first fundamental frequency where λ 1 = Equation 2 also holds true for multispan beams with pinned intermediate supports but the length is treated as the effective length of the beam between each span. The λ term is again numerically determined and varies depending on the number of spans in the system and the mode of interest. Belvins provides exact solutions to the frequency parameter, λ, for beams with an axial load under several boundary conditions other than the free-free condition and a few others. Further, Mukhopadhyay [8] gives the equation for the fundamental frequency of a pinned-pinned beam with an axial load in Eq. 3 but no equation is presented for the beam with an axial load and the free-free boundary conditions. f 1 = π 2 EI ρal 4 (1 ± PPcr ), (3) where ρ is the mass density, A is the cross-sectional area, and P cr is the Euler (critical) buckling load of the beam. For the cases without exact solutions, Belvins presents the following approximation, Eq. 4: ( f i P 0 = 1 + P f i P =0 P cr λ 2 1 λ 2 i ) 1/2 i = 1, 2, 3,... (4) Continuing with focused work on the effect of axial on beams, Shaker [9] presents a detailed derivation of the beam equation with an axial load for various boundary

22 9 conditions. The derived equation of motion for the beam with a compressive axial load is given as Eq. 5: EI 4 w(x, t) x 4 + P 2 w(x, t) x 2 + µ 2 w(x, t) t 2 = 0, (5) where x and t are the spatial and temporal variables of the beam and w is the displacement in the direction normal to the x axis. Without the presence of the axial load the boundary conditions of a free-free beam are zero shear and moment. Shaker shows the effect of the axial load on the boundary conditions where the moment is still zero but the shear becomes Eq. 6: { } d 3 w dw + k2 = 0 dx3 dx x=0 { } d 3 w dw + k2 = 0, (6) dx3 dx x=1 where k is the load parameter defined as Eq. 7: k 2 = P EI. (7) Working through the derivation Shaker provides the characteristic equation for the beam with a compressive axial load as Eq. 8: [ 2β 6 (1 cos α 2 cosh α 1 ) + k 2 (k 4 + 3β 4 ) sin α 2 sinh α 1 ] = 0. (8) Additionally, Galef [10], Pilkington [11], and Bokaian [12] provide further background and results of axially loaded beams. As the characteristic equation (8) can not be solved for a closed form solution, numerical methods must be used to determine the non-dimensional frequency value, β. Discussing an effective method for determining β, Liu [13] used computer software

23 10 to determine the values of β and allow for a range of tensions to be evaluated. Lui further discusses the steps taken towards approximating equations that can be used to solve for β. String-Beam Systems Attempts to model one-dimensional architectures consisting of the payloadsupport interaction represented with a beam-string model yielded several examples of previous models. A variety of work has been done on the cable-stayed beam structures by Gattulli [14] and [15] and others. Cable-stayed structures are discussed in relation to pretensioned structures by Jones et al. [16] and [17] as a means for providing additional stiffness to the structure. Models based strictly on the beam-string coupling produced little results. Cao and Zhang [18] and [19] produced work on the nonlinear dynamics of a beam-string model. While similar to the model discussed in Chapter 1, this work incorporates a harmonic loading and boundary conditions supported by springs (Fig. 7). The governing equation developed for Fig. 7 has many similarities to the equations of motion developed for the one-dimensional string-beam model. Using the same notation as Figure 7: Beam-String model developed for nonlinear analysis

24 11 the beam equation of motion Zhang gives Eq. 9: 2 w 1 µ 1 + EI 4 w 1 t 2 x + c w 1 4 t [ + T 0 + K s 2 L 0 { L ( ) 2 w1 dx x P 0 F 2 cos Ω 2 t + EA 2L 0 ( ) } 2 w2 dx] 2 w 1 x x = µ 1F 2 1 cos Ω 1 t, (9) where the material properties are defined as previously mentioned w 1 is the displacement of the beam and w 2 is the displacement of the string, Zhang also adds an axial harmonic excitation through P 0 and F 2, and a fundamental vibration to the system through F 1. Equation 9 can be simplified to the equation of motion Eq. 5, presented by Shaker by removing the damping term, c, the harmonic forcing functions, F 1 and F 2, and the nonlinear dynamic term for the tension. For the nonlinear dynamic analysis the tension in the string is said to vary with the deflection as a function of time but not position [20]. Treating the analysis as a linear dynamics problem removes term the following term from Eq. 9: L 0 ( ) 2 w1 dx. (10) x Equation 9 is shown as a method for verifying the approach used to construct the model in this thesis. The model developed by Zhang [18] goes on to represent the nonlinear dynamics of the model for the forced and harmonic loadings applied. The beam-string model was also used to model a fiber optics system (Cheng and Zu [20]). This model represents a fiber optic coupler where optic fibers are bonded to a substrate. Treating the substrate as a beam member and the fibers as a string member, the configuration can be seen in Fig. 8. While the the system doesn t incorporate the payload tension in the string member, it does incorporate the increased number of ties that will be investigated. Cheng and Zu work through both

25 12 G. Cheng, J.W. Zu / Journal of Sound and Vibration 268 (2003) y 1, y 2 String: y 2 Beam: y1 O x K L 0 l L 0 K Shock motion y s Fig. 2. A simplified model of an optical fiber coupler. Figure 8: Beam-String model developed for a fiber optic coupler linear and nonlinear dynamics of the problem and work to decouple the boundary conditions to focus on the string response. Deployable Structures In developing the design variables for monitoring the changes in the system behavior, an understanding of what goes into the design and development process is a must. Hedgepeth [21] presents a good overview of the requirements of Large Space Structures (LSS). Detailing stiffness and precision requirements, member slenderness, and design examples, Hedgepeth offers insight to the factors are to be considered for any quality spacecraft design. Lake et al. [22] brings the ideas presented by Hedgepeth towards the current design methods. Lake presents solutions to the fundamental issues that must be considered for the design of LSS. Murphey [23] discusses the deployment and structural architectures used in deployable structures. He introduces the fundamental principals of mast design through concepts of mass efficiencies and boom optimizations,among other things. Recently

26 13 work has been focused on increasing the understanding of the deployable structures. Mikulas et al. [3] focused on the tension stiffened architectures and the effects of packaging and deployment in one and two-dimension applications. Focusing specifically on the pretensioned structures, Jones et al. [16] began detailing the effects of pretension on a system relative to the mass ratios among other things. This system was followed with the presentation of the relationship between tension and the mass [17]. The results from Jones allowed for the development of the work herein by providing a basis to build on.

27 14 FINITE ELEMENT MODEL ABAQUS Support Model Setup To gain an understanding of how the system behaves, a simple model was designed through ABAQUS finite element modeling software. Working with free-free boundary conditions, the support only model and the support-payload model were analyzed and verified through comparisons with closed form solutions. The work presented in this chapter was started as a model based on fixed-free boundary conditions but as the work progressed this was seen to be an inaccurate representation of the supportpayload model. Working with the fixed-free boundary conditions would be looking at half of the structure where the payload and support would be attached to a base member. Working with structures on a large scale, possible over 100 meters, a base member would be very small relative to the system. As a result looking at the full scale of the model and treating the base member as part of the mass of the support yields a free-free system with more flexibility and an improved representation of a realistic model. Further, Hedgepath [21] notes that the assumption can be made that the flexible part of the structure will be the dominant part of the system and free-free boundary conditions are accurate. The work on the free-free support-payload model was started through the analysis of the support member alone. The support member was verified for frequency and buckling values through ABAQUS methods and compared with the relevant equations. The model was then modified to incorporate the payload member. Due to the complexity of the system no previously published work has addressed the analytical definition of this coupled system. As a result, a comparison was made with the work

28 15 published by Jones et al. [17] as noted in Chapter 2 and similar methods were used as Jones et al. presented on the finite element approach to this problem [16]. The beam members used herein fall into the category of a slender Euler beam and are based on the assumptions given for the Euler-Bernoulli beam theory [24]. The beams are uniform along the span and composed of a linear, homogeneous, and isotropic elastic material. The beam meets the necessary slenderness ratios where the cross sectional dimension is much less than the length of the beam or the distance between the connection points. At high values of the number of ties (i.e. as N approaches infinity) the slenderness ratio could pose issues with the accuracy of the model, but this work shows this issue is minimized as the solution converges towards solution regardless of the number of ties before the slenderness ratio is violated. The deformation is only considered in the normal direction to the beam axis and the transverse shear strain is neglected. Further the maximum load that can be applied to the beam is the Euler (critical) buckling load as defined in Eq. 11: P cr = π2 EI L 2 (11) From these beam assumptions the support only model was set up using 50 B21H elements. The B21H elements are hybrid two node beam elements which allow for displacement and rotation in two directions. The material properties were set for an arbitrary material type with the notion that the model will be non-dimensionalize and applicable for all materials and geometric properties. Material properties were specified through the *BEAM GENERAL SECTION command which required inputs for area, moment of inertia, density, and elastic modulus among others. To simulate the prestress loading in the payload a compression load (Fig. 9) was applied at each end node through the concentrated load command, *CLOAD. The

29 16 "#!! $# Figure 9: Model Representation of ABAQUS Support only model load value was applied over a range of values from zero to the critical buckling load. From this model two calculations were performed through two individual steps for frequency and buckling load. Each step was treated as a non-linear geometry to meet the criteria of ABAQUS. Initial models were run with linear geometries as this appeared to be the appropriate case for the system. After consulting the ABAQUS manuals it was determined that the non-linear geometry must be used to achieve the desired results for the frequency and buckling calculation steps. The ABAQUS code code for the support model can be seen in Appendix A. The accuracy of the model is presented in Table 1. Equations 2 and 11 were used to calculate the numerical values for frequency and buckling load in Table 1. These values show an excellent correlation with the theoretical model as expected for such a simple model. These values become important in the coupled model as this beam only model becomes the bounding condition for the system. In-addition to verifying the frequency and buckling commands of ABAQUS, the prestress ability was also investigated and verified. The prestress command was com- Table 1: Verification of ABAQUS Beam Model Frequency (Hz) Buckling Load (kn) Calculated Numerical % error

30 Theoretical 5.0 Numerical Analytical 4.0 Frequency (Hz) Load Ratio, P/P cr Figure 10: Comparison and validation of ABAQUS approach pared against the compressive axial load to ensure similar results were achieved. As previously noted an increasing axial compressive force acting on a beam results in a decreasing frequency. To compare the axial loading to the prestress loading a range of loads from zero to the critical buckling load were analyzed. These values were further compared against the approximate solution shown in Eq. 4 and the beam solution derived by Shaker [9] in Eq. 8. Figure 10 shows how close each method is to the exact solution presented by the Theoretical curve. ABAQUS Support-Payload Model Setup To increase the complexity of the model, the payload member was added as a truss system with truss elements connected to the beam elements of the support at the ends of the beam. Connecting the payload and support at the ends corresponds to the N = 2 case for the number of ties. Figures 11 and 12 show the general setup

31 18 for the support-payload model. Figure 11 uses the previous notation for showing the interaction between the support and payload where Fig. 12 further shows the interaction between the payload and support as represented by the beam and truss elements. With the inclusion of the payload, the compressive force previously acting on the support by the axial load or prestress force was converted to a prestress tension in the payload. This prestress value was calculated based on the load of interested and the area of the payload. To represent the payload as a string both truss elements and beam elements were considered. Very similar results could be obtained by using a truss element incapable of carrying a bending stiffness or using a beam element with a reduced bending stiffness (EI). Ultimately the truss element was used but the same results could have been obtained with the beam elements. The payload was created using T2D2H truss elements with geometric properties governing the area, elastic modulus, Poisson s ratio, and density. The geometric properties were defined with the *SOLID SECTION command. The area of both the payload and support were set to the same value for the model to allow for simple manipulation of the mass as suggested by Jones [16]. By adjusting the density of both components the desired mass ratios could be obtained. Using different areas for 78!*+,-*.(01231$!! 98!! "566-$#(01231$ Figure 11: General Representation of Support-Payload model for N=2

32 19 78 "#$%&'()!*+,-*./(01231$ 98 41*2()"566-$#/(01231$ Figure 12: ABAQUS representation of the interaction between the payload and support for N=2 the components greatly increased the time to determine the component and system masses. The payload and support were joined with element ties at the desired location. The nodes at the tie location were set using the *SURFACE, TYPE=NODE command and tied together with the *TIE command. Again using the *INITIAL CONDITIONS, TYPE=STRESS command to set the prestress in the payload, the system frequency could be calculated. An initial step was run to equalize the prestress and a second step was run to calculate the display the first several natural frequencies. The ABAQUS code for the support-payload model can be seen in Appendix A. Payload Support Model for N greater than 2 The model used for the base N=2 configuration was easily modified for an increased number of ties. For N = 3, with an added connection point at the middle of the beam, the ABAQUS model incorporates one more tie location. The remainder of the input file is unchanged. The N=3 model can be seen for the general description in Figure 13 and as modeled in ABAQUS in Fig. 14. As the number of ties is increased

33 20 the tie locations are set to ensure a symmetrical system. This helps to keep the system simple and solvable. Any model configuration can then be set by creating these ties at the necessary locations. To create the flexural architecture system the nodes could be redfined to encompass both the payload and the support and thus be tied at every locations. As previously mention, the tie spacing needs to be set such that the slenderness ratio between the tie points and the cross-section of the beam remains suitable. 78!*+,-*.(01231$!! 98!! "566-$#(01231$ Figure 13: The general support-payload model for N = 3 78 "#$%&'()!*+,-*./(01231$ 98 41*2()"566-$#/(01231$ Figure 14: The representation of the ABAQUS support-payload model for N = 3

34 21 MATHEMATICAL MODEL Model Development Working with the model setup developed through the finite element model a mathematical approach was taken to define the analytical relationship between the payload and support members. This approach looks at the known derivations and equations of beam and string members and attempts to combine these members and create a system equation that is representative of the model and compares to the numerical results. As shown in Figs. 12 and 14, the model is treated as the interaction between a beam and a string. This gives two independent equations coupled through the boundary conditions of the payload. Three approaches are taken to capture this interaction and attempt to produce a useful method of determining the system frequency. The first approach simply solves the coupled equations for the frequency of the system. The second approach uses a transformation of coordinates to decouple the equations and solve them independently before combining the equations to obtain a system equation for the frequency. The last approach treats the system as an interaction between two beam members in an attempt to further capture the contributions of the payload. Jones [16] presented the ability to treat the system either as a beam-string system or a beam-beam system. To treat the payload as a beam member a reduction of the bending stiffness relative to the support bending stiffness must be used to minimize the flexural contribution from the payload.

35 22 Coupled Beam-Sting Equations The first approach to a system equations uses the coupled beam-string equations to find the system frequency. The free-free boundary conditions of the system allow for the beam equation of to be solved independent of the string equation. The solution is then used in the coupled boundary conditions of the string to solve for a system equation and frequency. Support Equation For the simple configuration shown in Fig. 12 the system is stiffened through the tenison in the payload, this can be treated as a compressive force in the support. Taking a differential element of the beam from Fig. 12 the governing equation as well as the boundary conditions can be solved by summing the forces and moments. The differential element under small rotation is shown in Fig. 15. Figure 15: Forces and moments acting on a differential element of the beam

36 23 To begin the derivation summing forces, where Q is the shear force, M is the bending moment, and dx is the differential length, and simplifying yields Eq. 12: Q + dq ρadx 2 z t Q 2 = 0 dq dx z ρa 2 t 2 = 0 (12) Summing moments and simplifying yields Eq. 13: M M + dm P dz Qdx = 0 dm dx P dz dx Q = 0 (13) Let w be the displacement in the z -direction, i.e., w = 0 + z = z. Based on standard definitions the moment can be defined as Eq. 14: EI d2 w = M (14) dx 2 Combining Eq. 14 with Eq. 13 and solving for Q yields Eq. 15: d dx (EI d2 w dw ) P dx 2 dx Q = 0 (EI d3 w dw + P dx 3 dx ) = Q (15)

37 24 Equation 15 can by combined with Eq. 12 to create the equation of motion, where µ s is the support mass per unit length which gives Eq. 17: x ( EI d3 w dw P dx 3 dx ) µ 2 w s t 2 = 0 EI 4 w x + P 2 w 4 x + µ 2 w 2 s t 2 = 0 (16) 4 w x + P 2 w 4 EI x + µ s 2 w 2 EI t 2 = 0 (17) Since w is a function of x and t, the solution can be solved using separation of variables. The time component can be assumed to hold the form in Eq. 18: T (t) = sin(ωt) (18) where ω is the natural frequency of the system. Substituting Eq. 18 into Eq. 17 eliminates the time component and can rewritten as, Eq. 19: d 4 w dx + P d 2 w 4 EI dx µ s 2 EI w = 0 (19) and simplified to Eq. 20: w iv s + k 2 w s β 4 w s = 0 (20) where k and β are defined in Eq. 21: k 2 = P (EI) s β 4 = µ sω 2 (EI) s (21)

38 25 The problem can be non-dimensionalized by setting the following variables and constants as Eq. 22: x = x L w = w L k = k L β = β L (22) Equations 20 and 21 can then be written in the non-dimensional forms of Eqs. 23 and 24: w iv s + k 2 w s β 4 w s = 0 (23) k 2 = P L2 (EI) s β 4 = µ sω 2 L 4 (EI) s (24) Boundary Conditions: For the free-free state, the boundary conditions are defined where the shear force and bending moment will be zero at the ends of the beam. These conditions are defined in Eqs. 14 and 15 and can be rewritten in nondimensional terms as in Eqs 25 and 26: M(x) = (EI) s L Q(x) = (EI) s L 3 d 2 w (25) dx( 2 ) d 3 w dw + k2 (26) dx3 dx Setting these equal to zero for the free-free condition and simplifying gives the boundary condition equations as Eqs. 27 and 28: d 2 w s dx 2 = 0 x=0 d 2 w s dx 2 = 0 (27) x=1 { } d 3 w s dx + dw 3 k2 s = 0 dx x=0 { } d 3 w s dx + dw 3 k2 s = 0 (28) dx x=1

39 26 Support Solution: Then the assumed solution to the equation of motion for the support, Eq. 23, as a function of x is given as Eq. 29: w s (x) = A cosh(α 1 x) + B sinh(α 1 x) + D cos(α 2 x) + E sin(α 2 x) (29) where constants are defined as Eq. 30: α 1 = ( k 2 /2 + k 4 /4 + β 4) 1/2 α 2 = ( k 2 /2 + k 4 /4 + β 4) 1/2 (30) The boundary conditions at x = 0 can be used to eliminate constants D and E in Eq. 29 and gives Eqs. 31 and 32: D = A α2 1 α 2 2 (31) E = B α3 1 + k 2 α 1 α 3 2 k 2 α 2 = B α 2 α 1 (32) The solution for the support structure can now be reduced to Eq. 33 w s (x) = A{cosh(α 1 x) + α2 1 cos(α α2 2 2 x)} + B{sinh(α 1 x) + α 2 sin(α 2 x)} (33) α 1 The solution for w s and the alpha term s are expand and shown in Appedix B. Payload Equation Solving for the equation of the motion of the payload can be done in a similar fashion to that of the support. For tensioned stiffened architecture the payload member contributes no bending stiffness and as noted is represented as a string.

40 27 Using the same method and giving the string a tension load which matches the compressive load applied to the support. This gives the equation of motion for the payload as Eq. 71: P 2 wp x 2 2 wp x 2 + µ 2 wp p + µ p P t 2 = 0 (34) 2 wp t 2 = 0 (35) Eliminating the time component by assuming that w p = w p (x) sin(ω p t) where ω p is the natural frequency of the payload and simplifying gives Eq. 36: w p ω2 a 2 w p = 0 (36) where a 2 = P µ p (37) Then the non-dimensional payload equations become Eqs. 38 w p γ 2 w p = 0 (38) γ 2 = ω2 a 2 (39) Payload Boundary Conditions: The payload is connected to the support structure at each end and therefore must have the same displacement. This gives the boundary conditions for the payload given in Eq. 40 as: w s (0) = w p (0) w s (1) = w p (1) (40)

41 28 Payload Solution The assumed solution to the equation of motion for the payload(eq. 36) is given for the payload structure (Eq. 41) as: w p (x) = F cos(γx) + G sin(γx) (41) where the constant γ is defined as: γ = ω/a (42) The payload equations can be non-dimensionalized by setting the variables and constants as Eq. 43: x = x L w p = w p L a = a L γ = γ (43) System Coupling: Now using the remaining boundary conditions for the support structure at x = 1 and the boundary conditions for the payload, 4 equations with 4 unknowns can be written to couple the payload and support equations. These 4 equations can then be used to determine the characteristic equation of the system. These equations are given in Eqs : C 1 {α 3 1 cosh(α 1 ) α 3 1 cos(α 2 )} + C 2 {α 3 1 sinh(α 1 ) α 3 2 sin(α 2 )} = 0 (44) C 1 {α2 3 sinh(α 1 ) + α1 3 sin(α 2 )} + C 2 {α2 3 cosh(α 1 ) α2 3 cos(α 2 )} = 0 (45) ( ) C α2 1 F = 0 (46) α2 2 { } { C 1 cosh α 1 + α2 1 cos(α α2 2 2 ) + C 2 sinh(α 1 ) + α } 2 sin(α 2 ) α 1 F cos(γ) G sin(γ) = 0 (47)

42 29 Isolating the coefficient matrix gives: α 1 3 cosh(α 1 ) α1 3 cos(α 2 ) α1 3 sinh(α 1 ) α2 3 sin(α 2 ) 0 0 α2 3 sinh(α 1 ) + α1 3 sin(α 2 ) α2 3 cosh(α 1 ) α2 3 cos(α 2 ) α α 2 2 cosh α 1 + α2 1 cos(α α 2 2 ) sinh(α 1 ) + α 2 2 α 1 sin(α 2 ) cos(γ) sin(γ) (48) The determinant of the coefficients yields Eq. 49: [ (α 3 1 sinh α 1 α 3 2 sin α 2 )(α 3 2 sinh α 1 + α 3 1 sin α 2 )( sin γ) (α 3 1 cosh α 1 α 3 1 cos α 2 )(α 3 2 cosh α 1 α 3 2 cos α 2 )( sin γ) ] = 0. (49) System Solution The equilibrium equation can be simplified to: sin γ [ 2α 3 1α 3 2(1 cos α 2 cosh α 1 ) (α 6 1 α 6 2) sin α 2 sinh α 1 ] = 0, (50) or more conveniently: sin γ [ 2β 6 (1 cos α 2 cosh α 1 ) + k 2 (k 4 + 3β 4 ) sin α 2 sinh α 1 ] = 0. (51) It should be noted that Eq. 51 is the characteristic equation of a beam with a compressive axial load with the sin γ term out front.

43 30 Decoupled Boundary Conditions The second approach to developing an analytical solution for the one-dimensional architectures is to decouple the boundary conditions between the payload and the support. The coupled payload and support equations as previously derived are shown in Eq. 23 for the support and Eq. 38 for the payload. w iv s + k 2 w s β 4 w s = 0 (23) w p γ 2 w p = 0 (38) These equations are coupled through the payload boundary conditions given in Eq. 75 and can be decoupled by transforming the coordinates as Eq. 52: z = w p w s. (52) Then the boundary conditions for the payload become Eq. 53: z(0) = 0 z(1) = 0 (53) Inserting Eq. 52 into Eq. 38 allows this decoupled boundary condition to be used, Eq. 54: d 2 (z + w s ) γ 2 (z + w dx 2 s ) = 0 ( ) d 2 z d 2 dx w s 2 γ2 z = dx 2 γ2 w s (54)

44 31 The solution for w s can be solved from the independent boundary conditions then inserted into Eq. 54 where a solution can be found for z. As solved the solutions for w s can be written as Eq. 55: w s (x) = C 1 sinh(α 1 x) + C 2 cosh(α 1 x) + C 3 sin(α 2 x) + C 4 cos(α 2 x) (55) where the constants have solved for as Eqs. 56: C 1 = 1 C 2 = α3 2 (cos(α 2 ) cosh(α 1 )) α2 3 sinh(α 1 ) + α1 3 sin(α 2 ) C 3 = α 2 C 4 = α2 1α 2 (cos(α 2 ) cosh(α 1 )) α 1 α2 3 sinh(α 1 ) + α1 3 sin(α 2 ) (56) The value for C 1 is set at 1 for approximation purposes. This value would have to be found experimentally to determine the amplitude of the system for a given forcing function. The mode shape produced can be seen to accurately approximate the free-free beam. With the known solution for w s, Eq. 54 becomes a non-homogenous ordinary differential equation and can be solved through the combination of a homogenous and particular solutions. To solve the homogenous equation the right hand side of Eq. 54 becomes zero, Eq. 57: d 2 z h dx 2 γ2 z h = 0 (57) This simple ordinary differential equation has a solution given in Eq. 58: z h (x) = Ae γx + Be γx (58)

45 32 Solving the particular solution can be found by expanding Eq. 55 and using the method of undetermined coefficients, where the particular component of the solution can be solved from Eq. 59: d 2 z p dx 2 γ2 z p = Φ(x) (59) The non-homogenous component can be simplified to Eq. 60 from the support solution. ( ) ( ) α Φ(x) = e α1x 2 (1 + C 2 ) 1 γ 2 α e α1x 2 ( 1 + C 2 ) 1 γ C 3 (α 2 2 γ 2 ) sin(α 2 x) + C 4 (α 2 2 γ 2 ) cos(α 2 x) (60) Setting up the solution, the particular component can be set as Eq. 61: z p = De α 1x + Ee αx + F sin(α 2 x) + G cos(α 2 x) (61) And the constants can be solved for as Eqs : D = 1 + C 2 2 E = 1 C 2 2 (62) (63) F = C 3 α 2 2 γ 2 α γ 2 = C 3 (64) G = C 4 α 2 2 γ 2 α γ 2 = C 4 (65) (66)

46 33 Further simplifying and converting back to the original form gives the solution for the particular solution as Eq. 67: z p (x) = sinh(α 1 x) C 2 cosh(α 1 x) + C 3 sin(α 2 x) + C 4 cos(α 2 x) (67) Then combining with the homogenous solution the equation for z becomes Eq. 68: z ( x) = Ae γx + Be γx sinh(α 1 x) C 2 cosh(α 1 x) + C 3 sin(α 2 x) + C 4 cos(α 2 x) (68) Although this solution doesn t solve for a system frequency, the individual frequencies found through this method offer insight into the behavior of the ABAQUS model. Beam-Beam Equation Derivation In addition to treating the payload as a string element, it might be possible to treat both payload and support as beams to ensure that both material properties are accounted for. Similar to ABAQUS you could treat the payload beam at much reduced material properties to negate the bending stiffness. This approach was verified as an appropriate assumption through numerical models. Beam-Beam Support Derivation The support could be treated with the same boundary conditions and compressive loading giving the same equations for the beam Eq. 23 and the boundary conditions Eqs. 27 and 28: w iv s + k 2 w s β 4 w s = 0 (23)