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DESIGN OF CLOSED LOOP DEPLOYA BLE STRUCTURES FOR TENTS AND MASTS By KARTHIGEYAN PUTHURLOGANATHAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004
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Copyright 2004 by Karthigeyan Puthurloganathan
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This thesis is dedicated to my father Mr. G.Loganathan, my mother Mrs. Sheela Loganathan and my brother Mr. P.L. Kamesh
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ACKNOWLEDGMENTS I would like to thank Dr. Carl D. Crane, III, the Director of the Center for Intelligent Machines and Robotics (CIMAR), for being my committee chairman and providing me with assistance and guidance throughout this work. I would also like to thank Dr. John Kenneth Schueller and Dr. Christopher Niezrecki for serving on my committee and offering their expertise. In addition I would like to thank Carol Chesney for her assistance towards completing this work. iv
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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................xii CHAPTER 1 DEPLOYABLE STRUCTURES..................................................................................1 1.1 Introduction.............................................................................................................1 1.2 Deployable Masts...................................................................................................2 1.2.1 Rod Controlled Deployment of Eight Stage Mast........................................3 1.2.2 Coilable Mast with Rotating Nut..................................................................4 1.2.3 Able Deployed Articulated Mast (ADAM)..................................................5 1.3 Deployable Tents....................................................................................................5 1.3.1 Rapidly Deployable Bennett Tent................................................................5 1.3.2 DRASH Tactical Shelters.............................................................................6 1.3.3 Very Large Area Shelter...............................................................................7 2 DESIGN OF A DEPLOYABLE CLOSED LOOP STRUCTURE FOR COLLAPSIBLE TENTS..............................................................................................9 2.1 Description of the Structure....................................................................................9 2.1.1 Angular Columns........................................................................................10 2.1.2 Plates...........................................................................................................10 2.1.3 Compressible Struts....................................................................................11 2.2 Stages of Deployment...........................................................................................12 2.2.1 Stage 0........................................................................................................12 2.2.2 Stage 1........................................................................................................13 2.2.3 Stage 2........................................................................................................13 2.2.4 Stage 3........................................................................................................14 2.2.5 Stage 4........................................................................................................14 2.3 Principle of the Mechanism..................................................................................15 2.4 Design Problem Statement...................................................................................17 2.4.1 Minimum Stiffness and Deflection of the Spring.......................................19 v
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2.4.2 Instantaneous Time, Velocity and Acceleration with Respect to Position............................................................................................................20 2.4.3 Alteration of the Stiffness of the Springs...................................................22 2.4.4 Forces Acting on the Hinge........................................................................25 2.4.5 Velocity and Acceleration with Respect to Position when the Structure is Deployed in a Single Stage...........................................................26 2.4.6 Stiffness and Minimum Deflection of the Spring when an External Load is Applied................................................................................................26 2.5 Numerical Example..............................................................................................27 3 DESIGN OF A DEPLOYABLE CLOSED LOOP MAST.......................................35 3.1 Description of the Mast........................................................................................35 3.2 Stages of Deployment...........................................................................................36 3.2.1 Stage 0........................................................................................................36 3.2.2 Stage 1........................................................................................................36 3.2.3 Stage 2........................................................................................................37 3.3 Principle of the Mechanism..................................................................................37 3.4 Design Problem Statement...................................................................................40 3.4.1 Minimum Stiffness and Deflection of the Spring.......................................41 3.4.2 Instantaneous Time, Velocity and Acceleration with Respect to Position............................................................................................................42 3.4.3 Alteration of the Stiffness of the Springs...................................................45 3.4.4 Forces Acting on the Hinge........................................................................47 3.4.5 Velocity and Acceleration with Respect to Position when the Mast is Deployed in a Single Stage..........................................................................48 3.4.6 Stiffness and Minimum Deflection of the Spring when an External Load is Applied................................................................................................48 3.5 Numerical Example..............................................................................................49 4 STIFFNESS COMPARISON OF MASTS................................................................57 4.1 Design Description of the Masts...........................................................................57 4.1.1 Mast I..........................................................................................................57 4.1.2 Mast II........................................................................................................58 4.1.3 Deployable Mast.........................................................................................59 4.2 Determination of Stiffness....................................................................................60 4.3 ANSYS Modeling of the Masts............................................................................60 4.4 Results...................................................................................................................61 5 SUMMARY AND CONCLUSIONS.........................................................................62 APPENDIX A MATLAB PROGRAMS............................................................................................64 B FIGURES FROM ANSYS ANALYSIS....................................................................83 vi
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LIST OF REFERENCES...................................................................................................90 BIOGRAPHICAL SKETCH.............................................................................................91 vii
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LIST OF TABLES Table page 2-1 Time, displacement, velocity and acceleration at different positions of layer 1................................................................................................................28 2-2 Time, displacement, velocity and acceleration at different positions of layer 2................................................................................................................29 2-3 Force on the hinge at different positions of layer 1...............................................31 2-4 Force on the hinge at different positions of layer 2...............................................31 2-5 Time, displacement, velocity and acceleration at different positions of the tent for a single stage deployment for tent.......................................................32 3-1 Time, displacement, velocity and acceleration at different positions of level 1................................................................................................................51 3-2 Time, displacement, velocity and acceleration at different positions of level 2................................................................................................................51 3-3 Force on the hinge at different positions of level 1...............................................53 3-4 Force on the hinge at different positions of level 2...............................................53 3-5 Time, displacement, velocity and acceleration at different positions of the mast for a single stage deployment for mast....................................................54 4-1 Stiffness and mass comparison of three different masts........................................61 viii
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LIST OF FIGURES Figure page 1-1 Rod controlled deployment of eight stage mast.......................................................3 1-2 Coilable mast...........................................................................................................4 1-3 ADAM mast.............................................................................................................5 1-4 Rapidly deployable Bennett tent..............................................................................6 1-5 DRASH shelter........................................................................................................7 1-6 Very large area shelter.............................................................................................8 2-1 Eight layer closed loop deployable structure...........................................................9 2-2 Angular column with four hinges..........................................................................10 2-3 Plate with four hinges............................................................................................11 2-4 Compressible strut.................................................................................................12 2-5 Stage 0 for tent.......................................................................................................13 2-6 Stage 1 for tent.......................................................................................................13 2-7 Stage 2 for tent.......................................................................................................14 2-8 Stage 3 for tent.......................................................................................................14 2-9 Stage 4 for tent.......................................................................................................15 2-10 Combination of compressible struts and vertical members called a layer.............15 2-11 Length of the strut when fully deployed, stowed and free length of spring..........16 2-12 Typical displacement vs force in the x direction for tent.......................................17 2-13 Position, velocity and acceleration analysis during deployment of layer 1...........29 2-14 Position, velocity and acceleration analysis during deployment of layer 2...........30 ix
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2-15 Force on the hinge at every position during the deployment of layer 1................31 2-16 Force on the hinge at every position during the deployment of layer 2................32 2-17 Velocity comparison between single stage and stage wise deployment of tent.....................................................................................................................33 2-18 Acceleration comparison between single stage and stage wise deployment of tent.....................................................................................................................33 3-1 Two level closed loop deployable mast.................................................................35 3-2 Stage 0 for mast.....................................................................................................36 3-3 Stage 1 for mast.....................................................................................................37 3-4 Stage 2 for mast.....................................................................................................37 3-5 Combination of struts and plates, a single side of a level......................................38 3-6 Length of strut when fully deployed, stowed and free length of spring................39 3-7 Typical displacement vs. force in z direction for mast..........................................40 3-8 Position, velocity and acceleration analysis during deployment of level 1...........51 3-9 Position, velocity and acceleration analysis during deployment of level 2...........52 3-10 Force on the hinge at every position during the deployment of level 1.................53 3-11 Force on the hinge at every position during the deployment of level 2.................54 3-12 Velocity comparison between single stage and stage wise deployment of mast....................................................................................................................55 3-13 Acceleration comparison between single stage and stage wise deployment of mast....................................................................................................................55 4-1 Mast I.....................................................................................................................57 4-2 Mast II....................................................................................................................58 4-3 Deployable mast.....................................................................................................59 B-1 Stiffness of mast I in the x direction......................................................................83 B-2 Stiffness of mast I in the y direction......................................................................84 B-3 Stiffness of mast I in the z direction......................................................................84 x
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B-4 Stiffness of mast I during torsion...........................................................................85 B-5 Stiffness of mast II in the x direction.....................................................................85 B-6 Stiffness of mast II in y direction...........................................................................86 B-7 Stiffness of mast II in z direction...........................................................................86 B-8 Stiffness of mast II during torsion.........................................................................87 B-9 Stiffness of deployable mast in x direction............................................................87 B-10 Stiffness of deployable mast in y direction............................................................88 B-11 Stiffness of deployable mast in z direction............................................................88 B-12 Stiffness of the deployable mast during torsion.....................................................89 xi
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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 CLOSED LOOP DEPLOYABLE STRUCTURES FOR TENTS AND MASTS By Karthigeyan Puthurloganathan May 2004 Chairman: Carl D. Crane, III Major Department: Mechanical and Aerospace Engineering This thesis presents the description of construction, stages of deployment, working principle and the mechanism design of closed loop deployable structures for collapsible tents and masts. Deployable structures are structures capable of large configuration changes. Deployable structures are usually used for easy storage and transportation as they are very compact when stowed and can be deployed into their service configuration as and when required. The structures designed in this thesis are purely spring driven. The structures are partially self deployable, completely stowable and very compact when stowed with minimum space utilization. The position, velocity and acceleration analysis of both the proposed deployable structures is performed. The stiffness of the proposed deployable mast is also compared with two other existing contemporary masts. The structures designed in this thesis consist of only three parts which are compressible struts, plates and hinges. These structures are very simple to construct, service and transport. xii
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CHAPTER 1 DEPLOYABLE STRUCTURES 1.1 Introduction Deployable structures are structures capable of large configuration changes. Most common is that the configuration changes from a packaged, compact state to a deployed, large state. These structures are usually used for easy storage and transportation. They are deployed into their service configuration when required. A well known example is the umbrella. Deployable structures are sometimes known under other names such as expandable, extendible, developable, and unfurlable structures. Deployable structures are both man-made and found in nature. Examples of naturally found deployable structures are tubeworms such as Sebella, sea anemones such as Metridium as well as leaves and flowers that deploy and fold on a daily basis. Man-made deployable structures have many potential applications both on earth and in space. In civil engineering, temporary or emergency structures have been used for a long time. A more recent application is the retractable roof of large sports stadiums. In space, deployable structures have been used since the former Soviet Union launched its first satellite, Sputnik, on October 4, 1957. Deployable structures have a wide variety of applications in space, because of their relative ease of storage, in booms, aerials and masts as well as in deployable solar arrays. The requirements that have to be met by deployable structures in their operational configuration (e.g., providing shelter from rain, in the case of umbrella, or forming an accurate reflective surface, in the case of deployable reflector antenna for 1
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2 telecommunications) are different from the requirements in the packaged configuration (usually, these should be as small as possible). But an essential requirement is that the transformation process should be possible without any damage, and should be autonomous and reliable [1]. The most obvious advantages of deployable structures are their optimization of space and mass when stowed. Other more indirect advantages of deployable structures are their ability to withstand high loads in the folded position. The ability of deployable structures to be made in-situ allows a single pass in manufacturing architectural structures such as domes. This is especially useful in space since on-site construction of erectable structures is tedious as well as risky. Potential disadvantages involved in using deployable structures include the trade-off between the size of the packaged structure and its precision in the deployed state. Both aspects are usually critical to the mission performance, but are sometimes conflicting requirements. The flexibility and ability of deployable structures to transmit random vibrations are hard to account for analytically. Hence deployable structures prove to be poor substitutes for carefully designed factory products. Some significant applications of deployable structures in the aerospace industry are masts, antennas, and solar panels. Some significant applications of deployable structures in the civil engineering field are tents for emergency housing or temporary shelters and as domes for sports stadiums. This thesis focuses on the development of closed loop self-deployable structures for tents and masts. 1.2 Deployable Masts Masts are suitable for most applications requiring the use of tall stable structures to provide secure support to antennas or any other equipment needed at specific height.
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3 Masts are most commonly used in microwave communications to elevate signals over any obstructions. Masts can be designed in a number of ways depending on the application. The most significant advantage of the masts discussed in this chapter over telescoping masts is that they are self deployable. 1.2.1 Rod Controlled Deployment of Eight Stage Mast This mast developed by Watt and Pellegrino [2], as shown in Figure 1-1, has a diameter of 0.23 m, height of 1.27 m and 0.27 m long struts made of 0.006 m diameter aluminum tubes connected by a pair of 0.019 m wide tape springs. Figure 1-1. Rod controlled deployment of eight stage mast
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4 In the stowed position, the mast was stored in a canister to prevent the struts from snapping back into their straight configuration. An aluminum rod running in the centre of the demonstrator mast was used to activate and control the deployment. The aluminum rod could be substituted with a storable tubular extendible member (STEM) or an inflatable tube. The complete, sequential deployment of the eight stage mast is shown in Figure 1-1. 1.2.2 Coilable Mast with Rotating Nut This mast developed by AEC-Able Engineering Company [3] has a higher stiffness, structural efficiency, and precision when compared to many other similar masts. This mast ranges from diameters of 0.15-0.5 m and lengths of 1-100 m. This mast is extremely light and is stowed at 2% of its deployed length. This mast involves an inexpensive method of controlled deployment, in which the tip mounted lanyard controls the deployment with a damper or motor and the boom tip rotates during deployment as shown in Figure 1-2. Figure 1-2. Coilable mast
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5 1.2.3 Able Deployed Articulated Mast (ADAM) This mast developed by AEC-Able Engineering Company [3] is used for applications requiring very long and stiff masts. This mast enables precision payload deployed operation without active controls and tailorable deployed properties to meet mission needs. This mast is compact and has a highly efficient stowage volume. ADAM is shown in Figure 1-3. Figure 1-3. ADAM mast 1.3 Deployable Tents Deployable tents, shelters have a wide range of applications in the military, aid organizations, temporary structures and leisure activities. Deployable tents are lightweight structures which are stowed and transported in a very compact fashion. They can also be deployed in a very short time with less manpower. A few examples of existing deployable tents are discussed below 1.3.1 Rapidly Deployable Bennett Tent The Bennett mechanism is applied by a team of Oxford inventors to develop a lightweight and rapidly deployable tent for the use of military or aid organizations [4].
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6 This tent is transported by air on standard pallets in large numbers. This structure consists of a series of simple mechanical linkages and is easily expandable to provide temporary accommodation of high strength suitable for use in a wide range of climatic conditions as shown in Figure 1-4. This structure collapses flat, facilitating compact packing and easy transport. This structure comprises of identical bars and hinges permitting complete interchangeability of members, easy part fabrication and maintenance. This structure is constructed only with a single layer facilitating greater internal volume and can withstand damage to one or more structural elements, which are also removable from special access points. Figure 1-4. Rapidly deployable Bennett tent 1.3.2 DRASH Tactical Shelters DRASH shelters are manufactured by DHS Systems, DRASH is an acronym for Deployable Rapid Assembly Shelter [5]. DRASH features a pre-attached two cover design which allows the user to quickly deploy the shelter without attaching or detaching the covers before or after deployment. DRASH provides the military with a very reliable,
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7 soldier-friendly equipment which is light-weight, man-portable, rapidly deployable and air transportable with no tools or assembly needed. DRASH consist of no locking devices, center supports, poles and beams. DRASH is available in a wide range of sizes from small command posts to large tactical operation centers. A 5.48 m wide and 13.1 m long DRASH shelter is shown in Figure 1-5. Figure 1-5. DRASH shelter 1.3.3 Very Large Area Shelter This re-deployable, self-erect shelter is constructed from the Ably Diamond system of aluminum components [6], which facilitates modular construction of rugged, high-stability, large-area structures to precisely meet demanding applications. With clear spans of up to 35 m this flexible system is adaptable to many configurations. Erection of this structure is a fast unskilled task which does not require any handling of mechanical equipments. The fabric of this structure is fixed and tensioned simultaneously for increase ease and speed in deploying the structure as shown in Figure 1-6. The anchorage of this structure is not limited to hard foundations, concrete or prepared ground
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8 and the structure easily accommodates uneven surfaces. This structure is deployable under windy conditions and has low multiple re-deploying costs. This shelter is used as helicopter hangars, aircraft sunshades, maintenance workshops, stores and other re-locatable buildings. Figure 1-6. Very large area shelter
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CHAPTER 2 DESIGN OF A DEPLOYABLE CLOSED LOOP STRUCTURE FOR COLLAPSIBLE TENTS This chapter presents the description of construction, stages of deployment, working principle and the mechanism design of the proposed deployable closed loop structure for collapsible tents along with a numerical example for better understanding. Angular column plate strut hinge Figure 2-1. Eight layer closed loop deployable structure 2.1 Description of the Structure The structure consists of angular columns, plates, compressible struts, springs and hinges (Figure 2-1). The angular columns and plates are vertical members while the top 9
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10 and bottom members of the compressible strut are attached to the hinges on the vertical members and are always at an angle with respect to the vertical members. The hinges permit only a single degree of freedom for the compressible struts relative to the vertical members it is attached to. A layer is defined as a combination of two compressible struts along with two vertical members and four hinges. The vertical members of every layer at all times are shared by two other layers and the structure has a total of four times the number of layers on any side (N). 2.1.1 Angular Columns Figure 2-2. Angular column with four hinges Angular columns are L shaped solid members that determine the height of the structure (Figure 2-2). They house four hinges which are connected to the compressible struts. The number of angular columns in this closed loop deployable structure is always equal to the number of sides the structure has. Any one of the angular columns can be chosen as the fixed link of the structure. 2.1.2 Plates Plates are solid members equal to the height of an angular column (Figure 2-3). They also house four hinges to which the struts are connected. The number of plates (C)
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11 on each side in this closed loop deployable structure is always equal to one less than the number of layers on that side. An L shaped angular column comprises of two plates welded perpendicular to each other on any one edge along the length. Figure 2-3. Plate with four hinges 2.1.3 Compressible Struts The struts consist of two members, (Figure 2-4), connected by a prismatic joint. The top member is a solid rod with a circular plate at the bottom, while the bottom member is a tube with an inner diameter equal to that of the plate in the top member. The length of the bottom member is chosen to be equal to the height of the deployed structure. The top and bottom members are always connected even when the structure is fully deployed due to the presence of a hard stop at the top end of the bottom member. The bottom member also houses a spring which in turn is always at an angle with respect to the plates and columns.
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12 Figure 2-4. Compressible strut 2.2 Stages of Deployment This type of closed loop deployable structure has 4N layers and 2N stages of deployment, where N is the number of layers on one side. In every stage of deployment two corresponding layers on opposite sides of the structure are deployed together. Given below are the four stages of deployment for an eight layer closed loop deployable structure. The angular column marked with the color red is the chosen fixed link. 2.2.1 Stage 0 At this stage (Figure 2-5), the structure is completely stowed and the springs are at maximum deflection.
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13 Figure 2-5. Stage 0, completely stowed structure 2.2.2 Stage 1 At this stage (Figure 2-6) the first layer on two opposite sides, which are furthest from the fixed link (shown in red in the top view) are deployed. Figure 2-6. Stage 1 2.2.3 Stage 2 At this stage (Figure 2-7), the second layer on the two opposite sides are deployed, Now two sides of the closed loop deployable structure are completely deployed.
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14 Figure 2-7. Stage 2 2.2.4 Stage 3 At this stage (Figure 2-8), the first layers, corresponding to the layers deployed in stage one, are deployed. Figure 2-8. Stage 3 2.2.5 Stage 4 At this stage (Figure 2-9), the second two layers corresponding to the layers deployed in stage three are deployed, hence the closed loop structure is completely deployed.
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15 Figure 2-9. Stage 4 2.3 Principle of the Mechanism This is a purely spring driven mechanism. When the structure is completely stowed, the springs are fully compressed and exert forces on the hinges primarily in the y and negative y direction. The moment the struts reach the angle after which the structure is self deployable, i.e., self deployable angle ( d, where is shown in Figure 2-10.), the springs exert sufficient force in the x direction to move the angular columns and plates fully deploying the structure. The force that fully deploys the mechanism is proportional to the cosine of the angle (Figure 2-12). Figure 2-10. Combination of compressible struts and vertical members called a layer
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16 When stowed Fx = K ((Lmax Lmin) +dmin) cos max = 0 { max = 90} (2.1) During deployment Fx = K ((Lmax Li) +dmin) cos i (2.2) Fully deployed Fx = K (dmin) cos min (2.3) Figure 2-11. Length of the strut when fully deployed, stowed and free length of spring As shown in Figure 2-11, Lmax, Li, and Lmin are the lengths of the struts at various positions. dmin is the difference between the free length of the spring and minimum length of the strut (the amount that the spring is compressed when the strut is fully deployed) and max, i, and min are the angles of the strut at various positions. At any instant of time the distance traveled by the angular columns and plates (Xi) is not equal to the deflection of the spring and the springs are always at an angle with
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17 respect to the direction of motion. Both these factors together contribute to the non-linear behavior of the mechanism. According to Newtons second law of motion (F=ma), the force applied by the springs is always equal to the product of the mass (M) of the vertical member and its acceleration. K((Lmax Li) +dmin) cos i = 22)(tXi M (2.4) The force applied by the spring on the mass follows a curve as shown in Figure 2-12. Figure 2-12. Typical displacement vs force in the x direction 2.4 Design Problem Statement Given: Deployed length and height of the structure (LN,H) Mass of the vertical members, top and bottom members of the struts (Mp Mts Mbts ) The initial displacement after which the structure is self deployable (Xd) Co-efficient of friction ( ) between the vertical members and the surface they will be sliding on.
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18 Find: 2.4.1 The minimum required stiffness and deflection of the springs in the structure based on friction and weight consideration of the struts. 2.4.2 Determination of instantaneous time, velocity and acceleration with respect to position in every layer of the structure. The total time taken to deploy the structure was also determined. 2.4.3 Alteration of the stiffness of the springs in various layers to develop a uniform structure in which the time taken to deploy every layer remains a constant. 2.4.4 Determination of forces acting on the hinge at various stages of deployment. 2.4.5 Determination of instantaneous time, velocity and acceleration with respect to position when the structure is deployed in a single stage. 2.4.6 Determination of stiffness and deflection of the springs when an external load is applied For the given structure, Friction Force Ff = 9.81 (m/sec2) (Mp ) (2.5) Number of layers on each side N = HLN [must be an integer] (2.6) Number of plates on each side C = N + 1 (2.7) Weight of the top member Wts = 9.81 (m/sec2) (Mts) (2.8) Length of strut when deployed Lmax = ( 22 N LN + H2).5 (2.9) Length of the struts when stowed Lmin = H Length of strut at any position Xi Li = (Xi2 + H2).5 (2.10) Each angular column consists of two plates. The plates are perpendicularly to each other and welded at one of the edges along their length as shown in Figure 2-2. Hence for simplicity reasons an angular column is considered as two plates.
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19 2.4.1 Minimum Stiffness and Deflection of the Spring The springs closest to the fixed link have the highest stiffness value while the springs farthest from the fixed link have the least stiffness value. The springs closest to the fixed link have to move the maximum mass against friction hence they have the highest stiffness compared to the other layers (Figure2-6 to Figure2-10). For any layer, at any instant of position Xi Working deflection of the strut from its maximum length dw = (Lmax-Li) (2.11) Angle of Strut i = (tan-1 iXH ) (2.12) The minimum deflection of spring dmin is not taken into account in equation (2.11). Hence at any instant of position Xi, the deflection of spring (di) is di = dw + dmin (2.13) The force applied by the springs in the x direction at the point from which the structure is self deployable (Xd) must be sufficient in magnitude to overcome friction between the vertical members and ground. Also, for successful deployment, the force at the deployed state should be no less than the x direction force at position Xd. If this condition is not satisfied the springs will not be able to apply sufficient force in the x direction towards the end of deployment to fully deploy the structure. Hence the deflection (dmin) of the spring when the structure is fully deployed ( = 45) can be obtained from K dmin cos45 K ((Lmax Ld) + dmin) cos d dmin dddLL cos45coscos)(max (2.14)
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20 Where, Ld is the length of the strut and d is the angle of the strut at position Xd, (the location from which the device is self-deployable) For any layer j, where j = 1 to N, numbered from the layer farthest from the fixed link to the layer closest to the fixed link and at any instant Xi minimum stiffness of the spring (Kj) can be determined from (2Kj ((Lmax Li) + dmin) -2Wts cos (90i)) cos i = (.5C+j) Ff Kj = iiiitsfdLLWFjC cos))((2cos)90cos(2)5(.minmax (2.15) Kj is determined for every position from Xd to LN1 for small position intervals, where LN1 is the length of a single layer. The maximum value of Kj is taken to be the minimum required stiffness for that particular layer. Hence, the minimum required stiffness and deflection of the springs for the layer j are determined. The Matlab program PROG1T as shown in the appendix A, is used to perform this calculation. 2.4.2 Instantaneous Time, Velocity and Acceleration with Respect to Position The minimum required stiffness and the minimum required deflection of the springs are known from equations (2.14) and (2.15). In any layer (j), for any values of stiffness and deflection of springs above the minimum required values, the instantaneous time, velocity and acceleration with respect to position are calculated. The total time taken to deploy the layer is also determined. From the stages of deployment shown in Figure 2-6 to Figure 2-10, it can be concluded that the time taken to deploy the total structure is twice the time taken to deploy any one side of the structure as at each stage of deployment, a layer on one side and the corresponding layer on the opposite side are deployed simultaneously. Let X, V, A, t and AV be the values of displacement, velocity,
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21 acceleration, instantaneous time and average acceleration. For any layer the force equation is 2K ((Lmax Li) +dmin) cos i 2Wts cos (90i) cos i = 22)(tXi (Mp (.5C+j)) + (.5C+j) Ff (2.16) As mentioned above with regards to equation (2.4) the structure is non-linear and hence a simulation technique is adopted to determine the velocity and time at every instant of position. The instantaneous time, velocity and acceleration for all positions is then determined. At position X0 = Xd t0 (X0) = 0 V0 (X0) = 0 A0 (X0) = )5(.cos)90cos(2)5(.cos))((2minmaxjCMWFjCdLLKpdixdtsfxdxd At position X1 = X0 + .01 A1 (X1) = )5(.cos)90cos(2)5(.cos))((211min1maxjCMWFjCdLLKpixtsfxx A1V (X1) = .5(A0 (X0) + A1 (X1)) t1 (X1) = )(2))(04.)(()(115.1120000XAXAXVXVVV V1 (X1) = V0 (X0) + A1V (X1) t1 (X1) At position X2 = X1 + .01 A2 (X2) = )5(.cos)90cos(2)5(.cos))((222min2maxjCMWFjCdLLKpixtsfxx
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22 A2V (X2) = .5(A1 (X1) + A2 (X2)) t2 (X2) = )(2))(04.)(()(225.2221111XAXAXVXVVV V2 (X2) = V1 (X1) + A2V (X2) t2 (X2) At position Xn = X (n-1) +.01 An (Xn) = )5(.cos)90cos(2)5(.cos))((2minmaxjCMWFjCdLLKpixntsfxnxn (2.17) AnV (Xn) = .5(An-1 (Xn-1) + An (Xn)) (2.18) tn (Xn) = )(2))(04.)(()(5.21111nnvnnVnnnnXAXAXVXV (2.19) Vn (Xn) = Vn-1 (Xn-1) + AnV (Xn) tn (Xn) (2.20) Total time taken to deploy the layer, TT = (2.21) niit1 Total time taken to deploy the structure TTS = 2 1NjjTT (2.22) Hence the instantaneous time, velocity and acceleration with respect to position for any layer is determined. The total time taken to deploy the structure is also determined. The Matlab program PROG2T as shown in the appendix A, is used to perform these calculations. 2.4.3 Alteration of the Stiffness of the Springs From equation (2.21) the time taken to deploy the various layers on any side is determined. For reasons due to uniformity and better performance (more importantly in space applications) the stiffness of the springs should be designed in such a way that the
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23 time taken to fully deploy any layer in the structure remains a constant. This is an approximation method and the solution obtained is not accurate. Let the stiffness of the springs for any layer be Kj. During the deployment of a layer the time taken (ti) over any position interval Xi to Xi+1 is known. For some position interval Xi to Xi+1 the time taken (ti) would be the least when compared to the time taken by the layer to deploy through any other position interval. Let this value of time be assigned to the variable LT. Then, from equation (2.19) Time taken for the structure to deploy from X0 to X1 = LT l01 = t1 X1 to X2 = LT l12 = t2 X2 to X3 = LT l23 = t3 Xn-1 to Xn = LT l(n-1)n = tn (2.23) Where l01, l12, l23,..., l(n-1)n are variables. From equation (2.21), LT (l01+ l12 + l23 + l34 +. + l(n-1)n) = TT (2.24) LT (l01+ l12 + l23 + l34 ++ l(n-1)n) = GT (2.25) Where, GT is the new time within which the layer should completely deploy and LT is the new least time for some position interval Xi to Xi+1. In both equations (2.24) and (2.25), (l01 + l12 + l23 +l34 +. + l(n-1)n) remain the same since the time taken to deploy through any two corresponding position interval in two different layers is proportional. The pattern at which the instantaneous time changes at every position interval is always the same for layers with different spring stiffness. From equation (2.25)
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24 LT = )..........()1(34232301nnlllllGT (2.26) Hence, for the altered structure, Displacement from Xd to X1 t1 = LT* l01 V1 = 101tXX A1 = 101tVV Displacement from X1 to X2 t2 = LT* l12 V2 = 212tXX A2 = 212tVV Displacement from Xn-1 to Xn tn = LT* l(n-1)n (2.27) Vn = nnntXX1 (2.28) An = nnntVV1 (2.29) From equations (2.27), (2.28) and (2.29) the velocity, acceleration and instantaneous time at every position is determined. Hence for every position Xn, Ki = ixniitsfipdLLWjCFAjCM cos))((2cos)90cos(2)5(.))5(.(minmax
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25 Approximate altered stiffness of the spring Kj = 1niiKn (2.30) As this is not an accurate but an approximated solution, the value of Kj should be used in 2.42 and iterated until the accurate value is obtained. Similarly, the stiffness of all springs in the corresponding layers is determined. The Matlab program PROG3 as shown in the appendix A, is used to perform these calculations. 2.4.4 Forces Acting on the Hinge Forces on the hinge when the structure is fully deployed F = Kdmin FX = Kdmin cos45 FY = Kdmin sin45 Forces on the hinge when the structure is fully stowed F = K((Lmax Lmin) +dmin) FX = K((Lmax Lmin) +dmin) cos90 FY = K((Lmax Lmin) +dmin) sin90 Forces on the hinge during deployment F = K ((Lmax Li) +dmin) (2.31) FX = K((Lmax Li) +dmin) cos i (2.32) FY = K ((Lmax Li) +dmin) sin i (2.33) Where, FX and FY are the forces in the x and y direction as shown in Figure 2-10., while F is the resultant force. Li and i are determined from the equations (2.10) and (2.12)
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26 depending on the displacement. The Matlab program PROG4T as shown in the appendix A, is used to perform these calculations. 2.4.5 Velocity and Acceleration with Respect to Position when the Structure is Deployed in a Single Stage The instantaneous time with respect to position is the same for all the layers after the stiffness of the spring is determined as in 2.43. The total time taken to deploy the structure in one single stage is equal to the time taken to deploy a single layer, when the structure undergoes stage wise deployment. The velocity and acceleration of the last vertical member on any side is, Velocity VN = nnntXNXN)()(1 (2.34) Acceleration AN = nNNtVV1 (2.35) Where, Xn, Xn-1 and tn are determined from equations (2.19) and (2.20). From equations (2.34) and (2.35) it is evident that the velocity and acceleration of the last vertical member on any side is N times the velocity and acceleration when the structure is deployed stage wise. The Matlab program PROG5T as shown in the appendix A, is used to perform these calculations 2.4.6 Stiffness and Minimum Deflection of the Spring when an External Load is Applied Let Fapx be the external force applied in the negative x direction when the structure is fully deployed, hence when fully deployed the springs should apply a counter active force in the x direction equivalent to Fapx to prevent any change in the shape of the structure. Let, Fapx = 2Kj (dmin) cos45 (2.36)
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27 If Fapx is greater than Fapx then the values of Kj and dmin remains the same and if Fapx is smaller than Fapx the values of Kj and dmin should be accordingly altered. The maximum possible value for the minimum deflection of the spring is given as a percentage (P) of the working deflection of the spring dw. Hence dmin = P Lw Let, Fapx = 2Kj dmin cos45 (2.37) If Fapx is greater than Fapx the values of Kj and dmin may remain the same as in equation (2.37) or the value of dmin is further optimized to a lower value such that Fapx is equal to Fapx. If Fapx is smaller than Fapx then the value of Kj should be altered. 2Kj = mindFapx (2.38) Maximum deflection of the spring dmax = dmin + (Lmax-Lmin) (2.39) Total length of the spring LS = Lmin +dmin (2.40) Hence, the stiffness, minimum deflection, maximum deflection and the total length of the spring is determined. The Matlab program PROG1 as shown in the appendix A, is used to perform these calculations. 2.5 Numerical Example Given: Length of the structure when deployed LN = 2 m Height of the structure when deployed H = 1 m Mass of each plate MP = .2 kg Mass of each top member of the strut Mts = .05 kg Mass of each bottom member of the strut Mbs = .05 kg
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28 Displacement after which the structure is self-deployable Xd = .2 m Co-efficient of friction between the plate and ground = .1 Solution: From equations (2.5) to (2.9) Friction Force Ff = .1962 N Number of layers on each side N = 2 Number of plates C = 3 Weight of the top member Wts = .4905 N Total length of strut when deployed Lmax = 1.414 m 2.4.1: From equations (2.11) to (2.15) Minimum deflection of the springs dmin = .1514 m Minimum stiffness of spring in layer 1 K1 = 4.4 N/m Minimum stiffness of spring in layer 2 K2 = 5.28 N/m 2.4.2: For layer 1 From equations (2.17) to (2.22) Stiffness of the spring K = 5.0 N/m Table 2-1 Time, displacement, velocity and acceleration at different positions of layer 1 Position of the layer Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the self deployable position 0 .2 0 .0851 At the fully deployed position 3.389 1 .5148 .0183 At the maximum acceleration position 2.419 .59 .3546 .2326
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29 Figure 2-13. Position, velocity and acceleration analysis during deployment of layer 1 For layer 2 Stiffness of the spring K = 5.5 N/m Table 2-2 Time, displacement, velocity and acceleration at different positions of layer 2 Position of the layer Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the fully stowed position 0 0 0 0 At the fully deployed position 4.226 1 .4287 .0071 At the maximum acceleration position 3.1 .59 .3009 .1674 Total time taken to deploy the structure TTS = 15.21 sec
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30 Figure 2-14. Position, velocity and acceleration analysis during deployment of layer 2 2.4.3: GT = 3.369 sec From equations (2.23) to (2.29) LT =. 0233 sec X = 180.1 sec LT = .0187 sec Approximate altered stiffness Kj = 6.841 N/m At the approximate altered stiffness value 6.841 N/m the layer fully deploys at 3.262 seconds. After iteration process in 2.42 the accurate stiffness value to fully deploy the layer at 3.369 sec is determined to be 6.62 N/m. 2.4.4: From equations (2.31) to (2.33) For layer 1
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31 Table 2-3 Force on the hinge at different positions of layer 1 Position of the layer FX (N) FY (N) F (N) At the fully stowed position 0 2.827 2.827 At the fully deployed position .5353 .5353 .7570 At the maximum acceleration position 1.027 1.742 2.022 Figure 2-15. Force on the hinge at every position during the deployment of layer 1 In Figure 2-15, the blue graph is the force in the x direction, the red graph is the force in the y direction while the black graph is the resultant force. For layer 2 Table 2-4 Force on the hinge at different positions of layer 2 Position of the layer FX (N) FY (N) F (N) At the fully stowed position 0 3.11 3.11 At the fully deployed position .8327 .5888 .5888 At the maximum acceleration position 1.13 1.916 2.225
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32 Figure 2-16. Force on the hinge at every position during the deployment of layer 2 In Figure 2-16, the blue graph is the force in the x direction, the red graph is the force in the y direction while the black graph is the resultant force 2.4.5: From equations (2.34) and (2.35) Table 2-5 Time, displacement, velocity and acceleration at different positions of the tent for a single stage deployment for tent Position of the structure Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the fully stowed position 0 0 0 0 At the fully deployed position 3.389 2 1.029 .0366 At the maximum acceleration position 2.419 1.18 .7092 .4652 In Figure 2-17, the blue graph is the velocity of a single layer when the structure undergoes stage wise deployment while the red graph is the velocity of the last vertical member on any side during a single stage deployment. In Figure 2-18, the blue graph is the acceleration of a single layer when the structure undergoes stage wise deployment
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33 while the red graph is the acceleration of the last vertical member on any side during a single stage deployment Figure 2-17. Velocity comparison between single stage and stage wise deployment of tent Figure 2-18. Acceleration comparison between single stage and stage wise deployment of tent 2.4.6: From equations (2.36) to (2.40)
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34 External load Fapx = 50 N Minimum deflection of the springs (when P=.4) dmin = .1657 m Stiffness of spring in layer 1 K1 = 213.3 N/m Stiffness of spring in layer 2 K2 = 213.3 N/m Maximum deflection of the spring dmax =.5779 m Total length of the spring LS = 1.577 m
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CHAPTER 3 DESIGN OF A DEPLOYABLE CLOSED LOOP MAST This chapter presents the description of construction, stages of deployment, working principle and the mechanism design of the proposed deployable closed loop mast along with a numerical example for better understanding. Figure 3-1. Two level closed loop deployable mast 3.1 Description of the Mast The deployable mast comprises of rectangular plates and spring loaded compressible struts (Figure 3-1). A level is defined as a combination of six spring loaded struts and six plate, where in the struts are connected to the plates with a hinge, permitting only a single degree of freedom for the struts relative to the vertical members it is attached to. The struts are constructed in the same fashion as described in chapter 2, section 2.13. Each rectangular plate is welded to two other rectangular plates in two 35
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36 corners resembling the cross section of an equilateral triangle. The lower plates of the bottom most level is the fixed link of the mast. 3.2 Stages of Deployment This type of closed loop deployable mast has N levels and N stages of deployment. At every stage of deployment one complete level is deployed. Given below are the two stages of deployment for a two level closed loop deployable mast. The plate marked with the color red is the fixed link. 3.2.1 Stage 0 At this stage (Figure 3-2), the mast is completely stowed and the springs are at maximum deflection. Figure 3-2. Stage 0, completely stowed mast 3.2.2 Stage 1 At this stage (Figure 3-3) the first level, which is furthest from the fixed link (shown in red in the top view) is deployed.
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37 Figure 3-3. Stage 1 3.2.3 Stage 2 At this stage (Figure 3-4), the second level which belongs to the fixed link is deployed. Figure 3-4. Stage 2 3.3 Principle of the Mechanism This is a purely spring driven mechanism. When the mast is completely stowed, the springs are fully compressed and exert forces on the hinges primarily in the x and negative x direction (Figure 3-5). The moment the struts reach the angle after which the mast is self deployable i.e., self deployable angle ( d, where is shown in Figure 3-5),
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38 the springs exert sufficient force in the z direction to move the plates, fully deploying the mast. The force that fully deploys the mechanism is proportional to the cosine of the angle (Figure 3-7). Figure 3-5. Combination of struts and plates, a single side of a level When stowed Fz = 6K((Lmax Lmin) +dmin) cos max = 0 { max = 90} (3.1) During deployment Fz = 6K ((Lmax Li) +dmin) cos i (3.2) Fully deployed Fz = 6K(dmin) cos min (3.3) As shown in Figure 3-6, Lmax, Li and Lmin are the lengths of the struts at various positions. dmin is the difference between the free length of the spring and minimum length of the strut (the amount that the spring is compressed when the strut is fully deployed) and max, i, and min are the angles of the strut at various positions
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39 Figure 3-6. Length of strut when fully deployed, stowed and free length of spring At any instant of time the distance traveled by the plates (Zi) is not equal to the deflection of the spring and the springs are always at an angle with respect to the direction of motion. Both these factors together contribute to the non-linear behavior of the mechanism. According to Newtons second law of motion (F=ma), the force applied by the springs is always equal to the product of the mass (M) of the vertical member and its acceleration. K ((Lmax Li) +dmin) cos i = 22)(tZi M (3.4) The force applied by the spring on the mass follows a curve as shown in Figure 3-7.
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40 Figure 3-7. Typical displacement vs. force in z direction 3.4 Design Problem Statement Given: Deployed height of the mast (H) Length of the rectangular plates (L p) Height of every level (Hl) Mass of the plates, top and bottom members of compressible strut (Mp Mts Mbts ) Displacement after which the mast is self deployable (Zd) Find: 3.4.1 The minimum required stiffness and deflection of the springs in the mast based on the weight consideration of the struts and plates. 3.4.2 Determination of instantaneous time, velocity and acceleration with respect to position in every level of the mast. The total time taken to deploy the mast was determined. 3.4.3 Alteration of the stiffness of the springs in various levels to develop a uniform mast in which the time taken to deploy every level remains a constant. 3.4.4 Determination of forces acting on the hinge at various stages of deployment. 3.4.5 Determination of instantaneous time, velocity and acceleration with respect to position when the mast is deployed in a single stage. 3.4.6 Determination of stiffness and deflection of the springs when an external load is applied
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41 For the given mast, Number of levels N = lHH [must be an integer] (3.5) Number of rectangular plates C = 3(N + 1) (3.6) Weight of the strut Ws = 9.81 (m/sec2)(Mts + Mbts) (3.7) Weight of the plate Wp = 9.81 (m/sec2)Mp (3.8) Length of strut when deployed Lmax = ( 22 N H + LP 2).5 (3.9) Length of the strut when stowed Lmin = LP Length of the strut at any position Zi Li = (Zi2 + Hl 2).5 3.4.1 Minimum Stiffness and Deflection of the Spring The springs closest to the fixed link have the highest stiffness value while the springs farthest from the fixed link have the least stiffness value. The springs closest to the fixed link have to move the maximum mass against gravity hence they have the highest stiffness compared to the other levels (Figure3-2 to Figure3-4). For any level, at any instant of position Zi, where Hl is the height of every single level Working deflection of strut from its maximum length dw = (Lmax-Li) (3.10) Angle of Strut i = (tan-1 iXH ) (3.11) The minimum deflection of spring dmin, is not taken into account in equation (3.10). Hence at any instant of position Zi, the deflection of spring (di) is di = dw + dmin (3.12) For successful deployment the force in the z direction at the deployed state should not be less than the force applied by the springs in the z direction at the point from which
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42 the mast is self deployable (Zd). If this condition is not satisfied the springs will not be able to apply sufficient force in the z direction towards the end of deployment to fully deploy the mast. Hence the deflection (dmin) of the spring when the mast is fully deployed ( = 45) can be obtained from, Kdmin cos45 K ((Lmax Ld) +dmin) cos d dmin dddLL cos45coscos)(max (3.13) Where, Ld is the length of the strut and d is angle of the springs at position Zd (the location from which the device is self-deployable). For any level j, where j = 1 to N; numbered from the level farthest from the fixed link to the level closest to the fixed link, where Wts is the weight of the top member At any instant Zi the minimum stiffness of the spring (Kj) can be determined from, 6Kj ((Lmax Li) +dmin) cos d = 3Wp (j) + 6Ws (j-1) +6Wts cos (90d) cos i Kj = diidtsspdLLWjWjW cos))(((6cos)90cos(6)1(6)(3minmax (3.14) Kj is determined for every position from Zd to Hl for small position intervals, where Hl is the height of a single level. The maximum value of Kj is taken to be the minimum required stiffness for that particular level. Hence, the minimum required stiffness and deflection of the springs for the level j are determined. The Matlab program PROG1M as shown in the appendix A, is used to perform this calculation. 3.4.2 Instantaneous Time, Velocity and Acceleration with Respect to Position The minimum required stiffness and the minimum required deflection of the springs are known from equations (3.13) and (3.14). In any level (j), for any values of stiffness and deflection of springs above the minimum required values, the instantaneous
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43 time, velocity and acceleration with respect to position are calculated. The total time taken to deploy the level is also determined. From the stages of deployment shown in Figure 3-2 to Figure 3-4, it can be concluded that the time taken to deploy the total mast is sum of the time taken to deploy every level of the mast. Let Z, V, A, t and AV be the values of displacement, velocity, acceleration, instantaneous time and average acceleration. For any level (j) the force equation is 6K ((Lmax Li) +dmin) cos i -3Wp (j) 6Ws (j-1) -6Wtscos (90d) cos i = 22)(tXi (3Mp (j) + 6Ms (j-1)) (3.15) As mentioned above with regards to equation (3.4) the mast is non-linear and hence a simulation technique is adopted to determine the velocity and time at every instant of position. The instantaneous time, velocity and acceleration for all positions is then determined. At position Z0 = Zd t0 (Z0) = 0 V0 (Z0) = 0 A0 (Z0) = )1(6)(3)1(6)(3cos)90cos(6cos))((6000min0maxjMjMjWjWWdLLKspspZZtsZZ At position Z1 = Z0 + .01 A1 (Z1) = )1(6)(3)1(6)(3cos)90cos(6cos))((6111min1maxjMjMjWjWWdLLKspspZZtsZZ A1V (Z1) = .5(A0 (Z0) + A1 (Z1)) t1 (Z1) = )(2))(04.)(()(115.1120000ZAZAZVZVVV
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44 V1 (Z1) = V0 (Z0) + A1V (Z1) t1 (Z1) At position Z2 = Z1 + .01 A2 (Z2) = )1(6)(3)1(6)(3cos)90cos(6cos))((6222min2max jMjMjWjWWdLLKspspZZtsZZ A2V (Z2) = .5(A1 (Z1) + A2 (Z2)) t2 (Z2) = )(2))(04.)(()(225.2221111ZAZAZVZVVV V2 (Z2) = V1 (Z1) + A2V (Z2) t2 (Z2) At position Zn = Z (n-1) +.01 An (Zn) = )1(6)(3)1(6)(3cos)90cos(6cos))((6minmax jMjMjWjWWdLLKspspZnZntsZnZn (3.16) AnV (Zn) = .5(An-1 (Zn-1) + An (Zn)) (3.17) tn (Zn) = )(2))(04.)(()(5.21111nnvnnVnnnnZAZAZVZV (3.18) Vn (Zn) = Vn-1 (Zn-1) + AnV (Zn) tn (Zn) (3.19) Total time taken to deploy the level, TT = (3.20) niit1 Total time taken to deploy the mast TTS = 1NjjTT (3.21) Hence the instantaneous time, velocity and acceleration with respect to position for any level is determined. The total time taken to deploy the mast is also determined. The Matlab program PROG2M as shown in the appendix A, is used to perform this calculation.
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45 3.4.3 Alteration of the Stiffness of the Springs From equation (3.20) the time taken to deploy the various levels is determined. For reasons due to uniformity and better performance (more importantly in space applications) the stiffness of the springs should be designed in such a way that the time taken to fully deploy any level in the structure remains a constant. This is an approximation method and the solution obtained is not accurate. Let the stiffness of the springs for any level be Kj. During the deployment of a level the time taken (ti) over any position interval Zi to Zi+1 is known. For some position interval Zi to Zi+1 the time taken (ti) would be the least when compared to the time taken by the level to deploy through any other position interval. Let this value of time be assigned to the variable LT. Then, from equation (3.18) Time taken for the mast to deploy from Z0 to Z1 = LT l01 = t1 Z1 to Z2 = LT l12 = t2 Z2 to Z3 = LT l23 = t3 Zn-1 to Zn = LT l(n-1)n = tn (3.22) From equation (3.22), LT (l01+ l12 + l23 + l34 +.. + l(n-1)n) = TT (3.23) LT (l01+ l12 + l23 + l34 +. + l(n-1)n) = GT (3.24) Where, GT is the new time within which the level should completely deploy and LT is the new least time for some position interval Zi to Zi+1. In both equations (3.23) and (3.24), (l01 + l12 + l23 +l34 +. + l(n-1)n) remain the same since the time taken for any two corresponding position interval in two different levels is proportional. The
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46 pattern at which the instantaneous time changes at every position interval is always the same for levels with different spring stiffness. LT = )..........()1(34232301nnlllllGT (3.25) Hence, for the altered mast, Displacement from Zd to Z1 t1 = LT* l01 V1 = 101tZZ A1 = 101tVV Displacement from Z1 to Z2 t2 = LT* l12 V2 = 212tZZ A2 = 212tVV Displacement from Zn-1 to Zn tn = LT* l(n-1)n (3.26) Vn = nnntZZ1 (3.27) An = nnntVV1 (3.28) From equations (3.26), (3.27) and (3.28) the velocity, acceleration and instantaneous time at every position is determined. Hence for every position Zn,
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47 ZnZnspZnZntsnspdLLjWjWWAjMjM cos))((6)1(6)(3cos)90cos(6))1(6)(3(minmax Kn = Approximate altered stiffness of the spring Kj = 1niiKn (3.29) As this is not an accurate but an approximated solution, the value of Kj should be used in case 3.42 and iterated until the accurate value is obtained. Similarly, the stiffness of all springs in the corresponding levels is determined. The Matlab program PROG3M as shown in the appendix A, is used to perform this calculation. 3.4.4 Forces Acting on the Hinge Forces on the hinge when the mast is fully deployed F = Kdmin FX = Kdmin cos45 FZ = Kdmin sin45 Forces on the hinge when the mast is fully stowed F = K((Lmax Lmin) +dmin) FX = K((Lmax Lmin) +dmin) cos90 FZ = K((Lmax Lmin) +dmin) sin90 Forces on the hinge during deployment F = K((Lmax Li) +dmin) (3.30) FX = K((Lmax Li) +dmin) cos i (3.31) FZ = K ((Lmax Li) +dmin) sin i (3.32)
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48 Where, FX and FZ are the forces in the x and z direction while F is the resultant force. Li and i are determined from the equations (3.10) and (3.11) depending on the displacement. The Matlab program PROG4T as shown in the appendix A, is used to perform this calculation. 3.4.5 Velocity and Acceleration with Respect to Position when the Mast is Deployed in a Single Stage The instantaneous time with respect to position is the same for all the levels after the stiffness of the spring is designed as in Case 3.43. The total time taken to deploy the mast in one single stage is equal to the time taken to deploy a single level, when the mast undergoes stage wise deployment. The velocity and acceleration of the topmost platform is, Velocity VN = nnntZNZN)()(1 (3.33) Acceleration AN = nNNtVV1 (3.34) Where, Zn, Zn-1 and tn are determined from equations (3.18) and (3.19). From equations (3.33) and (3.34) it is evident that the velocity and acceleration of the last vertical member on any side is N times the velocity and acceleration when the mast is deployed stage wise. The Matlab program PROG5M as shown in the appendix A, is used to perform this calculation. 3.4.6 Stiffness and Minimum Deflection of the Spring when an External Load is Applied Let Fapx be the external force applied in the negative z direction when the mast is fully deployed, hence when fully deployed the springs should apply a counter active force in the z direction equivalent to Fapx to prevent any change in the shape of the mast.
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49 Let, Fapx = 6Kj (dmin) cos45 (3.35) If Fapx is greater than Fapx then the values of Kj and dmin remains the same and if Fapx is smaller than Fapx the values of Kj and dmin should be accordingly altered. The maximum possible value for the minimum deflection of the spring is given as a percentage (P) of the working deflection of the spring dw Hence dmin = P dw Let, Fapx = 6Kj dmin cos45 (3.36) If Fapx is greater than Fapx the values of Kj and dmin may remain the same as in equation (3.36) or the value of dmin is further optimized to a lower value such that Fapx is equal to Fapx. If Fapx is smaller than Fapx then the value of Kj should be altered. 6Kj = mindFapx (3.37) Maximum deflection of the spring, dmax = dmin + (Lmax-Lmin) (3.38) Total length of the spring, LS = Lmin +dmin (3.39) Hence, the stiffness, minimum deflection, maximum deflection and the total length of the spring is determined. The Matlab program PROG1M as shown in the appendix A, is used to perform this calculation. 3.5 Numerical Example Given: Height of the mast when deployed H = 2 m Length of rectangular plates LP = 1 m Height of every single level Hl = 1 m Mass of each plate MP = .2 kg
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50 Mass of each top member Mts = .05 kg Mass of each bottom member Mbs = .05 kg Displacement after which the mast is self-deployable Zd = .2 m Solution: From equations (3.5) to (3.9) Number of levels on each side N = 2 Number of plates C = 9 Weight of the top member of the strut Wts = .9810 kg Weight of the plate Wp = 1.962 kg Total length of strut when deployed Lmax = 1.414 m 3.4.1 From equations (3.13) and (3.14) Minimum deflection of the springs dmin = .1514 m Minimum stiffness of spring in level 1 K1 =11 N/m Minimum stiffness of spring in level 2 K2 = 28.6 N/m 3.4.2 From equations (3.16) to (3.21) For level 1 Stiffness of the spring K = 12.5 N/m
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51 Table 3-1 Time, displacement, velocity and acceleration at different positions of level 1 Position of the level Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the fully stowed position 0 0 0 0 At the fully deployed position .4982 1 3.9 1.119 At the maximum acceleration position .3764 .59 2.714 13.74 Figure 3-8. Position, velocity and acceleration analysis during deployment of level 1 For level 2 Stiffness of the spring K = 30 N/m Table 3-2 Time, displacement, velocity and acceleration at different positions of level 2 Position of the level Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the fully stowed position 0 0 0 0 At the fully deployed position .7075 1 3.25 .0781 At the maximum acceleration position .5659 .59 2.283 10.03
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52 Figure 3-9. Position, velocity and acceleration analysis during deployment of level 2 Total time taken to deploy the mast TTS = 1.2 sec 3.4.3 Altering the value of spring stiffness in the second level to fully deploy at .4986 sec From equations (3.22) to (3.28) LT =. 0031 sec X = 227.3 sec LT = .0022 sec Approximate altered stiffness Kj = 40.81 N/m At the approximate altered stiffness value 40.85 N/m the level fully deploys at .4186 sec. After iteration process in case 3.42 the accurate stiffness value to fully deploy the level at .4986 sec is determined to be 36.42 N/m. 3.4.4 From equations (3.30) to (3.32) For level 1
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53 Table 3-3 Force on the hinge at different positions of level 1 Position of the level FX (N) FZ (N) F (N) At the fully stowed position 7.069 0 7.069 At the fully deployed position 1.338 1.338 1.892 At the maximum acceleration position 4.355 2.569 5.056 Figure 3-10. Force on the hinge at every position during the deployment of level 1 In Figure 3-10, the blue graph is the force in the z direction, the red graph is the force in the x direction while the black graph is the resultant force. For level 2 Table 3-4 Force on the hinge at different positions of level 2 Position of the level FX (N) FZ (N) F (N) At the fully stowed position 16.96 0 16.96 At the fully deployed position 3.211 3.211 4.542 At the maximum acceleration position 10.45 6.166 12.13
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54 Figure 3-11. Force on the hinge at every position during the deployment of level 2 In Figure 3-11, the blue graph is the force in the z direction, the red graph is the force in the x direction while the black graph is the resultant force 3.4.5 From equations (3.33) and (3.34) Table 3-5 Time, displacement, velocity and acceleration at different positions of the mast for a single stage deployment for mast Position of the structure Time (sec) Displacement (m) Velocity (m/sec) Acceleration (m/sec2) At the fully stowed position 0 0 0 0 At the fully deployed position .4982 2 7.801 2.239 At the maximum acceleration position .3764 1.18 5.429 27.49 In Figure 3-12, the blue graph is the velocity of a single level when the mast undergoes stage wise deployment while the red graph is the velocity of the topmost platform of the mast during a single stage deployment.
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55 Figure 3-12. Velocity comparison between single stage and stage wise deployment of mast Figure 3-13. Acceleration comparison between single stage and stage wise deployment of mast In Figure 3-13, the blue graph is the acceleration of a single level when the mast undergoes stage wise deployment while the red graph is the acceleration of the topmost platform of the mast during a single stage deployment
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56 3.4 6 From equations (3.35) to (3.39) External force applied Fapx = 30 N Minimum deflection of the springs (when P=.4) dmin = .1657 m Stiffness of spring in level 1 K1 = 42.5 N/m Stiffness of spring in level 2 K2 = 42.5 N/m Maximum deflection of the spring dmax =.5779 m Total length of the spring LS = 1.577 m
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CHAPTER 4 STIFFNESS COMPARISON OF MASTS This chapter presents the design description, mass and stiffness comparison and finite element modeling using ANSYS 7.1 of three different masts including the deployable mast designed in chapter 3. 4.1 Design Description of the Masts 4.1.1 Mast I This is a typical mast based on the work done by Billy Derbes, Member AIAA [7]. This mast consists of two levels with the same configuration arranged one over another and is constructed with struts and ball joints (Figure 4-1). Figure 4-1. Mast I 57
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58 The top and bottom platform of every level is triangular, made up of three struts of equal length, fixed to one another. Every level consists of twelve struts, where the top and bottom platform consist of six struts. Three struts are vertically arranged linking the corresponding vertices of the triangular shaped platforms with ball joints. Three other struts are at an angle, linking two different vertices of corresponding sides of the top and bottom platform with ball joints. 4.1.2 Mast II This is a mast based on the optimized platform of Jaehoon Lee [8]. This mast, constructed with struts and ball joints, consists of two levels with the same configuration arranged one over another in a fashion where the top level is inverted with respect to the bottom level (Figure 4-2). Figure 4-2. Mast II
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59 The top and bottom platform of every level is triangular, made up of three struts fixed to one another. The top platform of the bottom level, which is also the bottom platform of the top level, is made up of struts half as long as the struts in the other two platforms. In a level, every vertex of the platform with a larger base is linked to the ends of one side of the smaller platform with two struts connected by ball joints. Every level consists of twelve struts, where the top and bottom platforms consist of six struts. Six other struts are at an angle, linking the top and bottom platforms. 4.1.3 Deployable Mast This is based on the mast discussed in chapter 3. This mast consists of two levels arranged one over another and is constructed with struts, hinges and rectangular plates (Figure 4-3). Figure 4-3. Deployable mast
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60 The top and bottom platform of every level is triangular, made up of three rectangular plates of equal length, fixed to one another. Every level consists of six struts and six rectangular plates, where in the struts are always at angle with respect to the plates. The struts are connected to the plates with hinges permitting only a single degree of freedom. In every level the struts link two different ends of corresponding rectangular plates in the top and bottom platform. 4.2 Determination of Stiffness The stiffness of the mast is determined by the application of forces on the three inner vertices of the topmost platform of the mast and the resulting displacement of the vertices. The forces are applied along the x, y and z axis and a torque is generated by applying forces on the inner vertices of the topmost triangular platform in a direction perpendicular to the central axis. For every mast, forces are applied on three points in the topmost platform and stiffness is measured as the ratio of the sum of total forces applied to the average of displacements at these three points. KX = xxf31 (4.1) Where, KX is the stiffness along the x-axis, fx is the force applied along the x-axis and x is the displacement along the x-axis. Similarly the stiffness along the other axes is determined. The torsional stiffness KTZ is determined as the ratio of the sum of all torque to the average rotation. 4.3 ANSYS Modeling of the Masts Every strut in all three masts is a cylindrical tubular member with an outer radius of 0.35 m and an inner radius of 0.30 m. The height of every level in all the masts is 4.9 m
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61 and the length of each side of every platform is 5.2 m. A force of 75000 N is applied along the requisite axis, on the vertices of the topmost platform. Every strut and plate in all three masts is made of aluminum. In mast I the length of every vertical strut is 4.9 m and the length of every angular strut is 7.14 m. In mast II the length of each side of the central platform is 2.6 m and the length of every strut connecting two platforms is 5.12 m. In the deployable mast the length of every strut is 6.78 m. The thickness of the plates used in the top and bottom platform of every level is 0.1 m. In the ANSYS model the struts of the deployable mast are considered to be rigid when fully deployed, which can be made possible by a mechanism involving spring loaded pins. 4.4 Results Table 4-1 Stiffness and mass comparison of three different masts KX (N/m) KY (N/m) KZ (N/m) KtZ (Nm/degree) Mass (kg) Mast I 1.18E+08 1.18E+08 2.13E+09 1.12E+07 2.94E+04 Mast II 7.65E+07 7.65E+07 3.20E+09 1.62E+07 2.65E+04 Deployable mast 1.54E+06 1.56E+06 7.27E+07 2.35E+06 3.42E+04 From Table 4-1, it is evident that the deployable mast designed in Chapter 3, is less stiff and weighs more when compared to the typical regular masts. The figures pertaining to the results shown in Table 4-1 which is obtained from ANSYS 7.1 is shown in the appendix B.
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CHAPTER 5 SUMMARY AND CONCLUSIONS The description of construction, stages of deployment, working principle and the mechanism design of the proposed deployable closed loop structure for collapsible tents and masts is explored in this thesis. The position, velocity and acceleration analysis of the structures are also done. The stiffness of the deployable mast designed in this thesis is compared with two other existing contemporary designs of masts. Both the tent and mast designed in this thesis are structures which are deployable and stowable a number of times with ease. The most significant advantage of the two proposed structures is that, in the stowed configuration though the springs have maximum potential energy, the structures are in an equilibrium position. This is because the direction of the force applied by the springs when the structures are stowed is perpendicular to the direction of deployment. The proposed design for deployable tents and masts deploy in a very short time interval, time taken to deploy any layer in the tent or any level in the mast is very small and this is further supported by the numerical examples shown in Chapter 2 and 3. These structures are purely spring driven mechanisms and the time taken by a spring to reach its equilibrium position from a deflected position is very short which accounts for the fast deploying nature of these structures. The tent and mast designed in this paper, when stowed are compactly packed, with minimum space utilization. These structures consist only of three parts which are plates, hinges and compressible struts. Hence they are easy to construct, service and transport. 62
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63 These structures are not fully self-deployable, a starting force should be applied so that the angle between the compressible struts and plates reach the self-deploying angle ( d ), i.e., the angle after which the structure is self deployable. The finite element analysis in ANSYS 7.1 is done for the deployable mast and compared with two other regular masts. The trade-off in using a deployable mast is that it is less stiff and weighs more when compared to other typical masts All facets of the deployable structures designed in this paper are not yet studied. Due to the non-linear characteristics, the behaviors of the structures when a damper is involved pose an interesting puzzle. A beneficial solution would be to fabricate a to-scale model of the structures and determine the type of damping and the damping value (C) by a trial and error method, so that the structures has a short settling time. Also the vibration characteristics of the system should be determined and considered. If the need arises, starter springs may be designed to make the structures completely self deployable, in which case the structures will not be in equilibrium position when completely stowed.
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APPENDIX A MATLAB PROGRAMS PROG1T %Calculation of minimum deflection and stiffness L = input ('Input the total deployed length of the structure '); H = input ('Input the height of the structure '); mc = input ('Mass of the plate used '); mts = input ('Mass of the top strut '); fxd = input ('Maximum external force applied on the structure in the (-x) direction '); mu = input ('co-efficient of friction between plate and ground '); xd = input ('Position of the plate after which the system is self deployable '); fr = mc 9.81 mu; N = L / H; C = N + 1; HY = ( 2 H ^2 ) ^.5; %Calculation of minimum deflection dely = (( HY (( xd ^2 ) + ( H ^2 )) ^.5) cos (( atan ( H / xd)))) / ( cos (( pi / 180 ) 45 ) ( cos (( atan ( H / xd ))))); xd(1) = xd %Calculation of minimum stiffness for i = 1:N for j = 2 :(( H xd ) / .01) xd( j ) = xd ( j 1) + .01; 64
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65 k( j ) = ((( .5 C + i ) fr ) + 2 mts 9.81 cos (( atan ( H / xd( j ))))) / ( 2 ((( HY (( xd (j) ^2) + (H ^2)) ^.5) + dely) cos (( atan ( H / xd( j )))))); end kk( i ) = max (k ( j )); A( I ) = ( .4 ( HY H )) kk( i ) .7071; end for i = 1 : N if (A( i ) > fxd) nk( i ) = k( i ); ndely ( i ) = ( .4 ( HY H )); elseif (A( i ) < fxd) nk ( i ) = .5 fxd / (( .4 (HY H)) .7071); ndely( i ) = ( .4 ( HY H)); YD( i ) = ndely ( i ) + ( HY H); LS ( i ) = HY + ndely ( i ); end end %Results disp ( 'The number of layers on each side ); disp ( N ) disp (' The minimum deflection of the spring when external load is not considered'); disp ( dely ) disp ('The minimum values of K for every layer when external load is not considered'); disp ( kk ) disp ('The minimum deflection of the spring when external load is considered'); disp ( ndely ) disp ('The minimum value of K for every layer when external load is considered'); disp ( nk ) disp ('The total deflection of spring when the system is fully stowed'); disp ( YD )
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66 disp ('The total length of spring '); disp ( LS ) PROG2T %Calculation of the total time taken to deploy a layer % User-defined inputs: LL = input ('Input the total deployed length of the structure '); L = input ('Input the height of the structure '); mc = input ('Mass of the plate used '); mts = input ('Mass of the top strut '); mu = input ('co-efficient of friction between plate and ground '); xd = input ('Position of the plate after which the system is self deployable '); k = input ('Stiffness of the spring '); jj =input ('The layer to which the spring belongs '); dely =input ('The minimum deflection of the spring '); % Calculation of variables used in the equations below N = LL / L; c = N + 1; h = ( 2 L ^2 ) ^.5; m = mc 9.81; dely = .1514; fr = m mu wts = mts 9.81; %Constants: x ( 1 ) = 0; a ( 1 ) = 0; t ( 1 ) = 0;
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67 v ( 1 ) = 0; % Initial conditions: % cos(90-theta)cos(theta)=.5 sin(2*theta) x ( 2 ) = xd; a ( 2 ) = (((((( 2 k ( h (((x ( 2 ) ^2) + ( L ^2)) ^.5) + dely ) ( cos (( atan ( L / x ( 2 )))))) ) (.5 c + jj ) fr) ( 2 wts .5 ( sin ((( 2 atan( L / x ( 2 ))))))))) / ( m ( .5 c + jj))); t ( 2 ) = 0.0001; av ( 2 ) = a ( 2 ) / 2; v ( 2 ) = av ( 2 ) t ( 2 ); T ( 1 ) = 0; T ( 2 ) = 0; for i = 3 :(((L-xd)/.01)+1) x ( i ) = x ( i 1) + .01; a ( i ) = ((((( 2 k ( h (((x ( i ) ^2) + (L ^2)) ^.5) + dely) ( cos (( atan ( L / x ( i )))))) ) (.5 c + jj) fr) ( 2 wts .5 ( sin ((( 2 atan ( L / x ( i )))))))) / ( m (.5 c + jj))); av ( i ) = (a ( i 1) + a ( i )) / 2; t ( i ) = ( -v ( i 1 ) + (( v ( i 1)) ^2 + ( 4 av ( i ) (x ( i ) x ( i 1)))) ^ .5) / ( 2 av ( i )); v ( i ) = v ( i 1 ) + av ( i ) t ( i ); T ( i ) = T ( i 1) + t ( i ); end xx ( 1 ) = 0; vv ( 1 ) = 0; xx ( 2 ) = 2 xd; vv ( 2 ) = 0 ; for j = 3 : ((( L x d) / .01) + 1)
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68 xx ( j ) = 2 x ( j ); tt ( j ) = t ( j ); vv ( j ) = (xx ( j ) xx ( j 1)) / tt ( j ); aa ( j ) = (vv ( j ) vv ( j 1 )) / tt ( j ); end % Results disp ('Maximum velocity of the layer'); disp (max ( v )) disp ('Maximum acceleration of the layer'); disp (max ( av )) disp ('Total time taken to deploy the layer'); disp ( T ) PROG3T %Altering the stiffness value % User-defined inputs: LL = input ('Input the total deployed length of the structure '); L = input ('Input the height of the structure '); mc = input ('Mass of the plate used '); mts = input ('Mass of the top strut '); mu = input ('co-efficient of friction between plate and ground '); xd = input ('Position of the plate after which the system is self deployable '); k = input ('Stiffness of the spring '); jj = input ('The layer to which the spring belongs '); dely = input ('The minimum deflection of the spring '); GT = input ( The time within which the layer should deploy ); % Calculation of variables used in the equations below
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69 N = LL / L; c = N + 1; h = ( 2 L ^2 ) ^.5; m = mc 9.81; dely = .1514; fr = m mu wts = mts 9.81; %Constants: x ( 1 ) = 0; a ( 1 ) = 0; t ( 1 ) = 0; v ( 1 ) = 0; % Initial conditions: % cos(90-theta)cos(theta)=.5 sin(2*theta) x ( 2 ) = xd; a ( 2 ) = (((((( 2 k ( h (((x ( 2 ) ^2) + ( L ^2)) ^.5) + dely ) ( cos (( atan ( L / x ( 2 )))))) ) (.5 c + jj ) fr) ( 2 wts .5 ( sin ((( 2 atan( L / x ( 2 ))))))))) / ( m ( .5 c + jj))); t ( 2 ) = 0.0001; av ( 2 ) = a ( 2 ) / 2; v ( 2 ) = av ( 2 ) t ( 2 ); T ( 1 ) = 0; T ( 2 ) = 0; for i = 3 :((( L xd) /.01) + 1) x ( i ) = x ( i 1) + .01;
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70 a ( i ) = ((((( 2 k ( h (((x ( i ) ^2) + (L ^2)) ^.5) + dely) ( cos (( atan ( L / x ( i )))))) ) (.5 c + jj) fr) ( 2 wts .5 ( sin ((( 2 atan ( L / x ( i )))))))) / ( m (.5 c + jj))); av ( i ) = (a ( i 1) + a ( i )) / 2; t ( i ) = ( -v ( i 1 ) + (( v ( i 1)) ^2 + ( 4 av ( i ) (x ( i ) x ( i 1)))) ^ .5) / ( 2 av ( i )); v ( i ) = v ( i 1 ) + av ( i ) t ( i ); T ( i ) = T ( i 1) + t ( i ); end % Calculation of stiffness mint = min ( t ( i )); X = T ( i ) / mint; mingt = GT / X; for j = 3 : ((( L xd) / .01 ) + 1) S(j) = t ( j ) / mint; b(j) = S ( j ) mingt; vg ( j ) = ( x ( j ) x ( j 1)) / b ( j ); ag ( j ) = (vg ( j ) vg ( j 1 )) / b ( j ); end % Stiffness calculation of the spring depending on acceleration (ag) AV ( 2 ) = 0; for q = 3: ((( L xd ) / .01) + 1) KKK ( q ) = 2 ((( h (( x ( q ) ^2) + (L ^2)) ^.5) + dely) ( cos (( atan ( L / x ( q )))))); KKKK(q) = (( .5 c + jj ) m ag ( q )) + (( .5 c + jj) fr) + (2 wts ( cos((( atan (L / x ( q ))))))); KKKKK ( q ) = KKKK ( q ) / KKK( q ); AV ( q ) = KKKKK( q ) + AV ( q 1 ) end SS = AV ( q ) / q
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71 disp ('The Altered stiffness of the spring is '); disp ( SS ) PROG4T %Forces at the hinges % User-defined inputs: LL = input ('Input the total deployed length of the structure '); L = input ('Input the height of the structure '); k = input ('Stiffness of the spring '); dely = input('The minimum deflection of the spring '); h = ( 2 L ^2) ^.5; % calculation of the forces in the x direction, y direction and the % resultant force x ( 1 ) = 0 ; for i = 2 :(( L / .01 ) + 1) x( i ) = x ( i 1 ) + .01; fx ( i ) = (( k ( h ((( x ( i ) ^2) + (L ^2)) ^.5) + dely)) ( cos (( atan ( L / x ( i )))))); fy ( i ) = (( k ( h ((( x ( i ) ^2) + (L ^2)) ^.5) + dely)) ( sin (( atan ( L / x( i )))))); f ( i ) = ( k ( h ((( x ( i ) ^2 ) + ( L ^2 )) ^.5) + dely )); end % Graphical representation of the results plot (x, fx) hold on plot (x, fy, 'r') hold on
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72 plot (x, f ,'black') PROG5T %Calculation of the total time taken to deploy a layer in a single stage % User-defined inputs: LL = input ('Input the total deployed length of the structure '); L = input ('Input the height of the structure '); mc = input ('Mass of the plate used '); mts = input ('Mass of the top strut '); mu = input ('co-efficient of friction between plate and ground '); xd = input ('Position of the plate after which the system is self deployable '); k = input ('Stiffness of the spring '); jj =input ('The layer to which the spring belongs '); dely =input ('The minimum deflection of the spring '); % Calculation of variables used in the equations below N = LL / L; c = N + 1; h = ( 2 L ^2 ) ^.5; m = mc 9.81; dely = .1514; fr = m mu wts = mts 9.81; %Constants: x ( 1 ) = 0; a ( 1 ) = 0; t ( 1 ) = 0; v ( 1 ) = 0;
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73 % Initial conditions: % cos(90-theta)cos(theta)=.5 sin(2*theta) x ( 2 ) = xd; a ( 2 ) = (((((( 2 k ( h (((x ( 2 ) ^2) + ( L ^2)) ^.5) + dely ) ( cos (( atan ( L / x ( 2 )))))) ) (.5 c + jj ) fr) ( 2 wts .5 ( sin ((( 2 atan( L / x ( 2 ))))))))) / ( m ( .5 c + jj))); t ( 2 ) = 0.0001; av ( 2 ) = a ( 2 ) / 2; v ( 2 ) = av ( 2 ) t ( 2 ); T ( 1 ) = 0; T ( 2 ) = 0; for i = 3 :(((L-xd)/.01)+1) x ( i ) = x ( i 1) + .01; a ( i ) = ((((( 2 k ( h (((x ( i ) ^2) + (L ^2)) ^.5) + dely) ( cos (( atan ( L / x ( i )))))) ) (.5 c + jj) fr) ( 2 wts .5 ( sin ((( 2 atan ( L / x ( i )))))))) / ( m (.5 c + jj))); av ( i ) = (a ( i 1) + a ( i )) / 2; t ( i ) = ( -v ( i 1 ) + (( v ( i 1)) ^2 + ( 4 av ( i ) (x ( i ) x ( i 1)))) ^ .5) / ( 2 av ( i )); v ( i ) = v ( i 1 ) + av ( i ) t ( i ); T ( i ) = T ( i 1) + t ( i ); end xx ( 1 ) = 0; vv ( 1 ) = 0; xx ( 2 ) = 2 xd; vv ( 2 ) = 0 ; for j = 3 : ((( L x d) / .01) + 1) xx ( j ) = 2 x ( j );
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74 tt ( j ) = t ( j ); vv ( j ) = (xx ( j ) xx ( j 1)) / tt ( j ); aa ( j ) = (vv ( j ) vv ( j 1 )) / tt ( j ); end %RESULTS plot (x, vv, 'r') hold on plot (x, v, 'b') disp ('Maximum velocity of the system'); disp (max (vv)) disp ('Maximum acceleration of the system'); disp (max(aa)) disp ('Total time taken to deploy the layer'); disp (T) PROG1M % Calculation of minimum deflection and minimum stiffness H = input ('Input the total deployed height of the structure '); L = input ('Input the length of each rectangular plate '); HL = input ('Input the height of every layer '); mp = input ('Mass of the plate used '); mts = input ('Mass of the top strut '); mbs = input ('Mass of the bottom strut '); fxd = input ('Maximum external force applied on the structure in the (-x) direction '); xd = input ('Position of the plate after which the system is self deployable '); N =H / HL; C =3 (N+1); HY = (2 L ^2) ^.5;
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75 % Calculation of minimum deflection dely = ((HY ((xd ^2) + (HL ^2)) ^.5) cos (( atan (HL/xd)))) / (cos ((pi/180) 45) (cos (( atan ( HL/xd ))))); xd (1)= xd; for i =1: N for j =2 :(( HL xd) / .01) xd(j) = xd (j 1) + .01; % Calculation of minimum stiffness k (j) =(( 3 (mp 9.81) (i)) + (6 ((mts + mbs) 9.81) (i 1)) + 6 mts 9.81 .5 sin ((2 atan (HL /xd (j) ))))/ (( 6 (((HY (( xd (j) ^2) + (HL ^2)) ^.5) + dely) cos (( atan (HL/xd(j))))))); end kk (i)= max (k(j)); A (i) = (.4 (HY HL)) kk(i) .7071; end for i =1: N if (A(i) > fxd) nk (i)=k (i); ndely (i) = (.4 (HY HL)); elseif (A (i)