space elevator

DESIGN OF SPACE ELEVATOR by Vijaya Kaithi, B.E.M.E. A Thesis In Mechanical Engineering Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE In MECHANICAL ENGINEERING Approved Dr. Seon Han Chairman of Committee Dr. Jharna Chaudhuri Dr. Stephen Ekwaro-Osire Fred Hartmeister Dean of the Graduate School December, 2008 Texas Tech University, Vijaya Kaithi, December 2008.   ii   ACKNOWLEDGMENTS First and foremost I would like to expre ss my deepest gratitude to my advisor Dr. Seon Han for her continuous guidance, motiv ation and support towards my research. The faith she had shown in me and my work for the past two and half years made me to work hard and achieve my academic goals. I would also like to than k my thesis committee members, Dr. Jharna Chaudhuri and Dr. Stephen Ekwaro-Osire for reviewing my work and their valuable suggestions. I would like to dedicate this thesis to my parents, Yadaia h Kaithi and Manemma Kaithi, without whom this thesis would not ha ve been possible. I w ould also like to thank all our family members and well wishers in I ndia for their support. I would like to thank all my friends in Lubbock for their support and words of encouragement. Texas Tech University, Vijaya Kaithi, December 2008.   iii   TABLE OF CONTENTS ACKNOWLEDGMENTS………………………………………………………………...ii ABSTRACT……………………………………………………………………………...vi LIST OF TABLES………………………………………………………………..…….viii LIST OF FIGURES…………………………………………...........................................ix CHAPTER 1. INTRODUCTION……………………………………………………………………...1 1.1 Introduction……………………………………………………………………1 1.2 Literature Review……………………………………………………………...2 1.3 Motivation……………………………………………………………………..3 1.4 Thesis Outline…………………………………………………………………4 2. LINEAR MATHEMATICAL M ODEL OF SPACE ELEVATOR…………………….5 2.1 Introduction……………………………………………………………………5 2.2 Pearson’s Model……………………………………………………………….5 2.3 Design Ideas…………………………………………………………………...8 2.3.1 Tapered Cross-section……………………………………………….8 2.3.2 Countermass with Tapered Cross-section………………………….10 2.4 Equations of Motion of Space Elevator- Linear Decoupled Model…………12 2.5 Natural Periods of the Space Elevator……………………………………….13 3. NON-LINEAR MATHEMATICAL MODEL OF SPACE ELEVATOR…………….23 3.1 Introduction………………………………………………………………..…23 3.2 Equations of Motion………………………………………………………....23 Texas Tech University, Vijaya Kaithi, December 2008.   iv    3.3 Static Solution in the Radial Direction………………………………………28 3.3.1 Cross-sectional Area………………………………………...……..28 3.3.2 Derivation of Countermass……………………………………...…31 4. TIDAL FORCES…………………………………………………………………...…40 4.1 Introduction…………………………………………………………………..40 4.2 Formation of Tidal Forces………………………………………………...…40 4.3 The Earth-Moon Co-ordinate System………………………………………..42 4.4 Equation of Tidal Forces due to the Moon…………………………………..44 4.5 Equation of Tidal Forces due to the Sun……………………………………..46 4.6 PSD Plots of the Tidal Forces………………………………………………..48 5. RESULTS AND DISCUSSION……………………………………………………...59 5.1 Introduction…………………………………………………………………..59 5.2 Design of the Space Elevator……………………………………………...…59 5.2.1 Maximum Amplitude Criterion…………………………………....59 5.2.2 Point Mass Criterion………………………………..…...…………61 5.2.3 Resonance……………………………………………….…………61 5.2.4 Dynamic Response of the Elevator……………………….………..63 5.2.5 Accessible Destinations…………………………………………....64 6. CONCLUSIONS AND FUTURE WORK……………………………………………85 REFERENCES……………………………………………………………………….….86 APPPENDIX A. MATLAB CODES…………………………………………………........……88 B. FINITE DIFFERENCE METHOD………………………….....................…111 Texas Tech University, Vijaya Kaithi, December 2008.   v   C. NUMERICAL ANALYSIS OF LI NEAR EQUATIONS OF MOTION........115 D. NUMERICAL ANALYSIS OF NONLINEAR EQUATIONS OF MOTION.........................................................................................................119 Texas Tech University, Vijaya Kaithi, December 2008.   vi   ABSTRACT A space elevator is a tall structure that st arts from the surface of the Earth and reaches out beyond geostationary orbit (GEO) in space. The space elevator could be used to deliver payloads to any Earth’s orbit or send them to any other planet in the solar system. Currently, rocket propulsion is the on ly system that can deliver payloads to various destinations in space. The space elev ator can provide easier, safer, faster and cheaper access to space compared to rocket propulsion system. To date, most of the designs have been based on the simple linear decoupled dynamic analysis of the space elevator. These simple linear models neglect the Coriolis acceleration, elongation, and the coupling be tween the longitudinal and transverse motions. The simple models are useful for calculating the natural frequencies of the elevator, but they cannot give the accurate dynamic response. A three dimensional nonlinearly coupled model that includes th e Coriolis acceleration, elongation, and the coupling between the longitudinal and transverse motions is used to accurately model the dynamic response of the elevator. It is found that the linear models signifi cantly underestimate th e response of the system. Therefore, a non-linearly coupled m odel must be used to accurately predict the dynamic response of the system. However, the frequency contents obtained from the linear models are similar to those identif ied from the nonlinea rly coupled models. Therefore, the linear models can be used to predict the natural fre quency of the system for preliminary designs. Two design cases have been proposed in th is thesis. These desi gns satisfy various design criteria such as accep table taper ratio, point mass, maximum tensile strength and maximum displacements. The design also takes care that the natural periods are sufficiently away from the tidal periods due to the Moon and the Sun. The first design case of the space elevator having a length of 66,000 km will be useful for reaching Texas Tech University, Vijaya Kaithi, December 2008.   vii   destinations within the Eart h’s orbit and the second desi gn case having a length of 91,000 km can also reach the destin ations in solar orbit. Texas Tech University, Vijaya Kaithi, December 2008.   viii   LIST OF TABLES 2.1 Taper ratio for different materials……………………………………………………16 4.1 Mass of the Moon, the Earth and the Sun……………………………………………50 4.2 Distance from the Earth to the Moon and the Sun…...................................................50 Texas Tech University, Vijaya Kaithi, December 2008.   ix   LIST OF FIGURES 2.1 Forces acting on an incremental element of length ݎ݀…...………………………… 17 2.2 Cross-sectional area at various positions of the elevator for different values of ݄.… 18 2.3 Relation between point mass and elevat or length for different values of ݄...……… 19 2.4 Variation of ߚ with length of the elevator…………………………………………...20 2.5 Natural periods of longitudinal vibr ation with length of the elevator……………….21 2.6 Natural periods of transverse vibration with length of the elevator………………….22 3.1 Co-ordinate system of the space elevator……………………………………………34 3.2 Displacements in non-linear model of the space elevator………………………..….35 3.3 Difference between Euler and Lagrangian co-ordinates……………………………..36 3.4 Variation in the cross-sectional area of the elevator with elongation factor……..…..37 3.5 Forces acting on an incremental element………………………………………….…38 3.6 Forces acting on the countermass…………………………………………...……….38 3.7 Variation in the point mass to cable mass ratio for different values of ܥ...…………39 4.1 Gravitational pull on the Earth by the Moon causing tidal bulges…………………..51 4.2 Displacement in the position of the Moon after 24 Hrs……………………………...51 4.3 Spring and Neap tides……………………………………………………….……….52 4.4 The Earth- Moon- Sun system……………………………………………………….52 4.5 Relation between different reference frames…………………………...……………53 4.6 The Earth- Moon co-ordinate syst em including the space elevator……………….…54 4.7 Differential accelerations at different positions of elevator due to the Moon…….…55 4.8 Differential accelerations at different positions of elevator due to the Sun………….56 4.9 PSD plot of differential accelerations due to the Moon……………………………...57 Texas Tech University, Vijaya Kaithi, December 2008.   x   4.10 PSD plot of differential accelerations due to the Sun………………………...…….58 5.1 Longitudinal displacements of the pointmass for various lengths of the elevator for linear model…………………..…………….…………….………………..…….…...66 5.2 Transverse displacements of the poinmass for various lengths of the elevator for linear model………….…………………………..…………………………………...67 5.3 Displacements in the axial, equatorial a nd meridional directions of the pointmass for nonlinear model……………………………………….…………....…….………….68 5.4 Longitudinal natural periods of first, second and third mode s of vibration predicted by the linear models for different lengths of the elevator indicating the regions of possible resonance…………………………..………………………………..…........69 5.5 Transverse natural periods of first, sec ond and third modes of vibration predicted by the linear models for different lengths of the elevator indica ting the regions of possible resonance…………….….…………..……………………………..…...........70 5.6 Maximum displacements of the pointma ss in the longitudinal and transverse directions for various lengths of the elevator…………...…………................………71 5.7 Point mass over cable mass ratio for various lengths of the elevator………………..72 5.8 PSD plots for the longitudinal response of the pointmass predicted by the linear and nonlinear models ……………………………………......……………………….…..73 5.9 PSD plots for the transverse response of the pointmass predicted by the linear and nonlinear models ………………………………………………..……………….…..74 5.10 PSD plots for the transverse (equatorial) response of the pointmass predicted by the linear and nonlinear models …..................................................................................75 5.11 Displacements of the pointmass in the l ongitudinal and transverse directions for L = 66,000 km predicted by the linear model……………………………...…....76 5.12 Velocities of the pointmass in the l ongitudinal and transverse directions for L = 66,000 km predicted by the linear model………….......……………...…....77 5.13 Displacements of the pointmass in the axial, equatorial and meridi onal directions for L = 66,000 km predicted by the nonlinear model……………...........……..……78 Texas Tech University, Vijaya Kaithi, December 2008.   xi   5.14 Velocities of the pointmass in the axia l, equatorial and meridional directions for L = 66,000 km predicted by the nonlinear model…….....…………...............…79 5.15 Displacements of the pointmass in the longi tudinal and transverse directions for L = 91,000 km predicted by the linear model………...………….………….......80 5.16 Velocities of the pointmass in the l ongitudinal and transverse directions for L = 91,000 km predicted by the linear model……………….......………...……81 5.17 Displacements of the pointmass in the axial, equatorial and meridion al directions for L = 91,000 km predicted by the nonlinear model…………….……....…...…….82 5.18 Velocities of the pointmass in the axia l, equatorial and meridional directions for L = 91,000 km predicted by the nonlinear model…………...............…...…..…83 5.19 Solar orbits accessible predicted by the line ar different lengths of the elevator…...84 B.1 Descritization of elevator into n number of elements………………………...……114 Texas Tech University, Vijaya Kaithi, December 2008. 1 CHAPTER 1 INTRODUCTION 1.1 Introduction Currently, rocket propulsion is the only system to deliver payloads to their destinations in space. A Space elevator provides an alternative means to rocket propulsion. Space elevator is a physical conne ction from a point on earth to a point beyond geostationary orbit (GEO) in space. Th e GEO is one of the geosynchronous orbits that is directly above the Earth’s equator with a period equal to the Earth's rotational period and an orbital eccentricity of approximately zero. The GEO is 42,164 km away from center of the earth. At the GEO, the Ea rth’s gravitational force and the centripetal acceleration are balanced so that an object can remain stationary with respect to the Earth. For an object on an orbit lower than the GE O, the object will have to rotate faster than the Earth to prevent itsel f from falling, and for an object on an orbit higher than the GEO, the object will have to rotate slower th an the Earth. Therefore, the portion of the space elevator below GEO experiences a net force toward the Earth, and the portion of the space elevator above GEO experiences a net force away from the Earth. The space elevator is a tension structure that is in unstable equilibrium about the GEO. Stability can be provided by putting more mass on the portion above the GEO and anchoring it to the ground. Mathematical details of the forces on the GEO are given in Chapter 2. A space elevator is preferred to rocket propulsion system because it can provide easier, safer, faster and cheaper access to space. Using the rocket propulsion system, it costs about $400 million to launch a satellite into geo synchronous orbit [1]. Space elevator could reduce this launch costs to as low as $10 per kg of material transported into space [2]. Not only does it provide a cheap er passage to space, space elevator is a much safer alternative to rockets. Un til now, NASA has launched 120 space shuttle Texas Tech University, Vijaya Kaithi, December 2008. 2 missions, and two missions, Challenger in 1986 and Columbia in 2003, resulted in catastrophic failures. This failure rate is c onsistent with the failure rate of 2% in unmanned rockets (such as missiles). W ith the space elevator, the dangerous accelerations at lift-off and reentry associated with rocket flights can be eliminated. With the introduction of a space elev ator, launching a satellite in to the Earth’s GEO becomes merely ascending along the elevator. The sp ace elevator could be used to deliver payloads to any of Earth’s orbi ts or send them to any other planet in the solar system. One of the major hurdles which stands in the way of the f easibility of a space elevator is the material required for its cons truction. No building material was found to be suitable for the construction of space elevator until 1991, when the first carbon nanotubes were made [3]. The high tensile strength and low density of the carbon nanotubes (CNT) make them perfectly suitable for space elevator construction. Since then a lot of research have been performed in the area of carbon nanot ubes. Recent developments in this area include the first measured values for the te nsile strength of CNT by Yu [4, 5]. Yu found measured tensile strengths ranging from 11 to 63 GPa for individual CNT. 1.2 Literature Review The space elevator concept was first pr oposed, about fifty years ago, by Russian Scientist Artsutanov [6]. His idea was to use a geo stationary sate llite as a base for constructing a space elevator. In 1966, ocea nographers at the Scripps Institute of Oceanography and Woods Hole Oceanographic Institution led by John Isaacs, provided the first engineering analysis of a space elevator [7]. They named the space elevator Sky Hook. At that time no material was ava ilable which could satisfy the strength requirements of their proposed sky hook. In 1975, Jerome Pearson published the most cited and extensive analysis of the Space Elevator concept [8]. In this paper he discussed the vibrational modes of the space elevator due to pay loads, gravity and tidal forces. Arthur C. Clar ke introduced this concept to a popular audience with his novel, “Fountains of Paradise” [9]. Texas Tech University, Vijaya Kaithi, December 2008. 3 Bradley Edwards conducted a feasibility analysis on th e space elevator in his report submitted to NASA under the NASA Ins titute for Advanced Concepts (NIAC) program [1]. This report presented a detailed study of the problems that would be encountered in designing, constructing, deploying and utilizing the space elevator. Edwards proposed a deployment strategy, consisting of sending a Spacecraft to GEO carrying the spooled cable [10]. Initially the cab le will be released in both upwards and downward directions with the help of two sp indles. The cable that is released in downward direction is pulled by gravity and it is anchored to the Earth after it is fully extended. The other cable that is released in the upward direction is pulled by outward centrifugal acceleration and after it reaches the desire d location, the Spacecraft at GEO moves on that cable to act as a countermass. Finally climbers with the cable will ascend and descend along this initial cable until a sufficiently thick string is created. 1.3 Motivation 1. So far, only Simple decoupled linear dynamic analysis of a space elevator is available. The simple model doesn’t take either the Coriolis force or th e coupling between the longitudinal and transverse motions into account. The simple m odel only provides the natural frequencies of the elevator, but cannot give an acceptable dynamic response. 2. To model the dynamic response accurately, a three dimensional nonlinearly coupled model that includes the Corio lis acceleration is needed. 3. The purpose of this thesis is to a. accurately model the dynamic respons e of the space elevator using a three- dimensional coupled model. b. propose feasible designs for the space elevator Texas Tech University, Vijaya Kaithi, December 2008. 4 1.4 Thesis Outline In Chapter 2, Pearson’s model is explained and two de sign modifications proposed by previous studies are also implemented. The first is to make the cross- sectional area of elevator tapered to maintain the constant stress level along the elevator. The second is to place a countermass at the tip of the elevator to decrease its length. Simple linear equations of moti on in the longitudinal and transv erse directions have been derived for the space elevator. Linear models are decoupled from each other, and the Coriolis accelerations, elonga tion, and the geometric coupling between the two motions are neglected. Natural periods of the space elevator in the longitudinal and transverse directions are calculated. In Chapter 3, the coupled and non-linear partial differential equations of motion for the three dimensional motion of the el evator are derived using the Hamilton’s principle. Using the static solution of th ese equations, significance of the elongation factor is explained. In Chapter 4, the tidal forces from the Moon and the Sun are obtained. Equations for these tidal accelerations are derived by de fining the co-ordinate systems for the Earth- Moon and the Earth-Sun systems. The forcing periods of the tidal forces are found from their power spectral density plots. In Chapter 5, response of the elevat or in the longitudinal and transverse directions under the influence of tidal forces is calculated by solving the linear equations numerically using the finite difference method first. The maximum possible displacements for the elevator are plotted for various lengths of elevator using the modal analysis. Using different desi gn criterion such as the maxi mum point mass that can be carried up, possible resonance and accessible des tinations for various lengths of elevator, two design cases have been proposed. These tw o design cases of the elevator will be serving two different purposes. Texas Tech University, Vijaya Kaithi, December 2008. 5 CHAPTER 2 LINEAR MATHEMATICAL MODEL OF SPACE ELEVATOR 2.1 Introduction In this chapter, we will start with the basic model of the space elevator proposed by Pearson [8]. Two design ideas have been proposed to Pearson’s model to maintain the lower constant stress level along the cable a nd to reduce the length of the elevator. Two- dimensional equations of motion for the sp ace elevator are derived by decoupling the longitudinal and transverse motion. These equatio ns have been numerically solved to find the natural periods of the first three modes of vibrat ion of elevator in the transverse and longitudinal directions for various lengths of the elevator. 2.2 Pearson’s Model Pearson’s version of space elevator is a long uniform struct ure having its base on the surface of the Earth and extending up to GEO [8]. Consider a small incremental element of the elevator of arbitrary length ݎ݀ ,which is located at a distance ݎ from the Earth’s center as shown in Figure 2.1 . The forces acting on the element are gravitational force acting towards the Earth and the centrip etal force acting away from the Earth. The elevator is located at the equator so that th e centripetal accelerati on acts along the cable. Let us consider an incremental element of mass dm as shown in Figure 2.1. The forces acting on this element are given by ՛ Σܨ ௥௔ௗ െ ܽ ݉݀ ௥, ௜௔௟ൌܨܶ݀ ௚ൌ ܶ݀ ൌܯܩ ௘݉݀ ݎଶെݒ ݉݀ଶ ݎ ሺ2.1ሻ where = Internal tension which varies along the elevator, ܶ݀ ܨ௚ = Gravitational pull given by ܨ௚ ൌ ܯ ܩ ௘ ݎ/݉݀ଶ, Texas Tech University, Vijaya Kaithi, December 2008. 6 = Radial acceleration given by ܽ௥ ൌ െ ݒଶ/r. ܽ௥ = Mass of the Earth, ܯ௘ ݉݀ = Mass of element = ݎ݀ ܣߩ , Density of the material used for elevator, ߩ = ܣ = Cross sectional area of the elevator, ܩ = Newton’s gravitational constant =6.67× 10ିଵଵ ே௠మ ௞௚మ , = distance from center of the Earth to the element of elevator, and ݎ ݒ = velocity of the elevator element due to the Earth’s rotation. The GEO is defined as the orbit where the gravitational force is balanced by the centripetal acceleration, or ܨ݀ ൌ 0 . The radial distance to the GEO is given by ݒ௦ଶൌܯܩ ௘ ݎ௦, ሺ2.2ሻ where = radius GEO = 42164 × 10ଷ ݉ , and ݎ௦ ݒ ௦ = velocity of elevator at GEO . Eqn 2.1 can be written as, ܶ݀ ൌ ቆܯܩ ௘ܣߩ ݎଶ െ ݒܣߩଶ ݎቇ .ݎ݀ሺ2.3ሻ Assuming that the el evator is stationary with resp ect to the Earth, the relationship between the velocity of elevator and its distance from the Earth is linear with the proportionality constant of ߗ .Therefore, we can write Texas Tech University, Vijaya Kaithi, December 2008. 7 ߗ ൌݒሺݎሻ ݎൌݒ௦ ݎ௦, ሺ2.4ሻ where ݒ ሻ is the velocity of elevator at a distance r from the center of the Earth and ሺݎ ߗ is the angular velocity of the Earth’s rotation and is given by ߗൌ2ߨ 24 60 ൈ 60ൈൌ7 . 2 7ൈ1 0ିହ ⁄ݏ݀ܽݎ The gravitational acceleration, ݃ ,is defined by ݃ ൌܯܩ ௘ ݎ଴ଶ , ሺ2.5ሻ where acceleration due to g n a f the Earth = 9.81 ݏ݉ ଶ⁄ and ݃ൌ ravity o surf ce o ݎ ଴ൌ radius of the Earth ൌ 6378 ൈ 10ଷ .݉ Substituting the Eqns 2.2, 2.4 and 2.5 into Eqn 2.3, the incremental force acting on the elevator becomes ܶ݀ ൌ ݎ݃ܣߩ ଴ଶ൬1 ݎଶെݎ ݎ௦ଷ൰ .ݎ݀ሺ2.6ሻ The tension at a distance ݎ can be obtained by integrating ܶ݀ fr m ݎ଴ ݎ or o to ܶ ሺݎሻൌܶሺݎ଴ሻെݎ݃ܣߩ ଴ଶቆ1 ݎ൅ݎଶ 2ݎ௦ଷቇ൅ݎ݃ܣߩ ଴ଶቆ1 ݎ଴൅ݎ଴ଶ 2ݎ௦ଷቇ, ሺ2.7ሻ where ܶሺݎ଴ሻ is the force at the ground to an chor the elevator to ground. Another important observation from E qn 2.7 is that the incremental force ܨ݀ on the incremental element, ݎ݀ ,is positive for ݎ൏ݎ ௦ and negative for ݎ ൐ ݎ ௦. That is, for the elevator to rotate at the Ea rth’s angular velocity about its own axis, an incremental element below GEO requires an incrementa l upward force and an incremental element Texas Tech University, Vijaya Kaithi, December 2008. 8 above GEO requires an incremental downward fo rce. Therefore, for the elevator to be self supporting, it must extend beyond GEO. The length of the self supporting to wer can be obtained by finding ݎ௙ such that ܨ൫ݎ ௙൯ൌ 0. Assuming that the force at the ground (to anchor the elevator to th e ground) is zero, we find that ݎ݃ܣߩ ଴ଶቆ1 ݎ଴൅ݎ଴ଶ 2ݎ௦ଷቇെݎ݃ܣߩ ଴ଶቆ1 ݎ௙൅ݎ௙ଶ 2ݎ௦ଷቇ ൌ 0, ሺ2.8ሻ the length of the elevator to be 144,000 km. 2.3 Design Ideas From previous studies, there are two desi gn ideas that we will utilize [8, 10]. The first is that the cross-sectiona l area should be tapered to main tain low constant stress level along the cable. The second is that the length of the elevator coul d be shortened if a counterweight is attached to the end of the elevator. In the following, we will discuss these two ideas. 2.3.1 Tapered Cross-section Let us consider a uniform tower (constant cross sectional area) such that the stress varies along the length of the tower and is given by ߪ ሺݎሻൌܶሺݎሻ ܣൌܶሺݎ ଴ሻ ܣെݎ݃ߩ ଴ଶቆ1 ݎ൅ݎଶ 2ݎ௦ଷቇ൅ݎ݃ߩ ଴ଶቆ1 ݎ଴൅ݎ଴ଶ 2ݎ௦ଷቇ. ሺ2.9ሻ If we assume that the force applied at the ground is zero, the location on the elevator which experiences the maxi mum stress could be found by equating ⁄ݎ݀ߪ݀ to zero. The value of ݎ found to be ݎ௦ and the corresponding value of ߪ is obtained by substituting ݎൌݎ ௦ in Eqn 2.9 or Texas Tech University, Vijaya Kaithi, December 2008. 9 ߪ ௠௔௫ ൌݎ݃ߩ ଴ଶቆ1 ݎ௢ ൅ݎ௢ଶ 2 ݎ௦ଷെ3 2ݎ௦ቇ ൌ ߩ ൈ 4.8424 ൈ 10଻ . ሺ2.10ሻ For steel, the maximum stress (using ߩ=7850 kg/m3) is about 358 GPa. The tensile strength of steel is typically about 250 MPa [11]. Therefore, it is impossible to build a uniform elevator using steel. For carbon nanotubes with density 1300 kg/m3, the maximum tensile strength is about 62.9 GPa, which is slightly under the experimentally measured strength of 63 GPa [4, 5]. The advantage of using a tapered cross-section is that it is theoretically possible to build an elevator using any material if we allow the cross-secti onal area to vary, by making the elevator thickest at GEO. However, as the allowable stress decreases, the taper ratio (defined by the ratio of the cross-sectional ar ea at GEO to that at the Earth’s surface) increases. For steel, we will show that the taper ratio is too large to be practical. The cross-sectional area as a function of the radial distance is obtained as follows. Starting with Equation 2.6, let us replace ܣ with ܣሺݎሻ. The goal is to obtain ܣሺݎሻ that will make the stress at any location constant, or ܨ݀ ൌ ܣ݀ ߪ . Then, we can write ܣ݀ ߪ ൌ ݎ݃ߩ ଴ଶ൬1 ݎଶെݎ ݎ௦ଷ൰ܣሺݎሻ ݎ݀ ܣ݀ ܣሺݎሻൌݎ଴ଶ ݄൬1 ݎଶെݎ ݎ௦ଷ൰ ,ݎ݀ሺ2.11 ሻ where ݄ൌ݃ߩ/ߪ is the specific strength of the elev ator and it has the units of length. By integrating the Eqn 2.11 from ݎ଴to ݎ, cross-sectional area can be written as ܣ ሺݎሻൌܣ ଴ ݁൭ቀ௥బ ௛ቁቀଵି௥బ ௥ቁା൬௥బమ ଶ௛௥ ೞయ൰൫௥ బమି௥మ൯൱ , ሺ2.12ሻ where ܣ଴ is the cross-sectional area of the tower at the surface of the Earth. Texas Tech University, Vijaya Kaithi, December 2008. 10 The above area equation in Eqn 2.12 is plotted (Assuming ܣ଴ൌ1) in Figure 2.2 with ݎ varying from ݎ଴ൌ 6378 km to 150,000 km . From figure 2.2, we can conclude that the cross-sectional ar ea of the elevator increases exponentially up to GEO and decreases exponentially from GEO towards the end. Ther efore the elevator will have maximum cross-sectional area at GEO. If we substitute ݎൌݎ ௦ in the Eqn 2.12, the taper ratio can be expressed as ܣ௦ ܣ଴ൌ݁ቀ௥బ ௛ቁ൬ଵା ௥బయ ଶ௥ೞయ ି ଷ௥బ ଶ௥ೞ൰. ሺ2.13ሻ If we substitute the corresponding values of ݎ଴ݎ ݀݊ܽ ௦ in Eqn 2.13, the taper ratio becomes ܣ௦ ܣ଴ൌ ݁ቀ଴.଻଻଺ ௥ బ ௛ቁ. ሺ2.14ሻ The taper ratio decreases with increasing ݄ .Theoretically, we can reduce the taper ratio by using material with high strength to weight ratio, which effectively increases ݄ , but there is a practical limit on the value of ݄ because of the upper limit on the stress that a certain material can endure. Another way to think about this is that we can reduce the stress by increasing the taper ra tio, so that any material can be used to make the space elevator if it is tapered. However, taper ratio for a material with low specific strength, ݄ , may be impractical. Taper ratio values for different materials ar e given in Table 2.1. Taper ratio for steel is so high that it is practically impossible to build an elevator. Theoretically, carbon nanotube can ha ve the tensile strength of 300 GPa. But the maximum measured value of tensile strength is 64.3 GPa. For this value of ߪ the corresponding taper ratio is 2.72, wh ich is quite possible. 2.3.2 Countermass with Tapered Cross-section Countermass is a lump of mass placed at th e tip of the elevator. The basic purpose of using countermass is to shorten the length of the elevator. By extending the elevator Texas Tech University, Vijaya Kaithi, December 2008. 11 beyond GEO, elevator experiences the requir ed centripetal force to balance the net weight of the tower. Instead of increasing the height of the elevator, we can put a lump of mass at a certain height above GEO such that it creates the centripeta l force required to keep the elevator in tension about GEO. The length of the elevator depends on the weight of the countermass, the relation between them is shown in Figure 2.3. Height of the elevator without countermass is 144,000 km [8]. The countermass is assumed as a point mass. The net force on the tower with length, ܮൌ ݎ ௙െݎ ଴, is given by ܹ ൌ ቆන ൬ܯܩ ௘ ݎଶെߗଶ ݎ൰݉݀௥೑ ௥బቇ൅ቆܯܩ ௘ܯ௣ ݎ௙ଶቇെ൫ ߗଶݎ௙ܯ௣൯. ሺ2.15ሻ In Eqn 2.15, the second term ீெ ೐ெ೛ ௥೑మ represents the gravitati onal force of attraction between countermass and the Earth and the third term ߗଶݎ௙ܯ௣ represents centripetal force acting on countermass due to the Earth’s rotation. The countermass that will make the net f ܹ ,zero is given by orce, ܯ ௣൫ݎ௙൯ൌቀ׬ቀܯܩ ௘ ݎଶെߗଶ ݎቁ ݉݀௥೑ ௥బቁ െቆܯܩ ݎ௙ଶቇ൅൫ ߗଶݎ௙൯ ൌ൬ܣ ߩ׬ሺݎሻݎ݃଴ଶ൬1 ݎଶെݎ ݎ௦ଷ ൰௥೑ ௥బݎ݀൰ െݎ݃଴ଶቆ1 ݎ௙ଶെݎ௙ ݎ௦ଷ ቇ. ሺ2.16ሻ The ratio of the point mass to the cable mass is given by ܯ௣൫ݎ௙൯ ݏݏܽ݉ ݈ܾ݁ܽܿൌ൬ܣ ߩ׬ሺݎሻݎ݃଴ଶ൬1 ݎଶെr ݎ௦ଷ ൰௥೑ ௥బݎ݀൰ െݎ݃଴ଶቆ1 ݎ௙ଶെr୤ ݎ௦ଷ ቇܣ ߩ׬ሺݎሻ௥೑ ௥బݎ݀ ሺ2.17ሻ Texas Tech University, aithi, December 2008. Vijaya K ൌ ൬ܣ׬ሺݎሻݎ݃଴ଶ൬1 ݎଶ 12 െr ݎ௦ଷ ൰௥೑ ௥బݎ݀൰ െݎ݃଴ଶቆ1 ݎ௙ଶെr୤ ݎ௦ଷ ቇܣ׬ሺݎሻ௥೑ ௥బݎ݀ , where ܣሺݎሻ is given by Equation 2.12. In Eqn. 2.17, point mass/cable mass will be a function of specific strength ݄ and length of the elevator ܮ .The relationship between the mass ratio and the length of the cable is shown in Figure 2.3. From Figure 2.3, we can observe that the Weight of the countermass needed increases with the decrease in length of the elev ator i.e. heavier mass is needed to shorten th e length of the elevator. 2.4 Equations of Motion of Space Elevator- Linear Decoupled Model As a first approximation, let us assume that the transverse motion and the longitudinal motion are decouple d. In addition, let us assume that the motions in either direction are small so that the linear mode ls are adequate. The transverse motion is modeled using the string model, and th e longitudinal motion is modeled using longitudinal beam model. The string model is valid for such a long and thin member where moment and shear are negligible [12]. The equation of motion is given by ݉ ሺݔሻ߲ଶݕ ݐ߲ଶൌ߲ܪ ݔ߲൬ܣሺݔሻݕ߲ ݔ߲൰൅݂ሺݐ,ݔ ሻ, 0 ൑ ݔ ൑ , ܮሺ2.18ሻ with boundary conditions given by ݕሺ0, ݐሻൌ0 , ܯ௣߲ଶݕ ݐ߲ଶቤ ௅,௧൅ܣܪ ሺܮሻݕ߲ ݔ߲ฬ ௅,௧ൌ In Eqn 2.18, ݉ሺݔሻ is mass per unit length, ݕሺݐ,ݔ ሻ is deflection of string and is function of both space and time. When the transverse motion is modeled, ݕሺݐ,ݔ ሻ and ܪ can be replaced by the transverse deflection, ݒሺݐ,ݔ ሻ , and tensile strength, .ߪ When the 0 . Texas ch longitudinal motion is modeled, ݕሺݐ,ݔ ሻ and ܪ can be replaced by the longitudinal deflection, ݑሺݐ,ݔ ሻ , and Young’s modulus, ܧ .Te University, Vijaya Kaithi, December 2008. 13 2.5 Natural Periods of the Space Elevator To find the natural frequencies of the elev ator, homogeneous part of the Eqn. 2.18 is considered and it can be written as, ߩ ܪݕሷ െܣᇱሺݔሻ ܣሺݔሻݕᇱሺݔሻെݕᇱᇱሺݔሻൌ 0. ሺ2.19ሻ In Eqn. 2.19, presents the derivative with respect to ݔ డ డ௫ ሺሻᇱ re ሺሻ ሺሻሶ represents the derivative with respect to time ݐ డ డ௧ሺሻ. Using the method of separation of variables, ݕሺݐ,ݔ ሻ can be expressed as ݕ ሺݐ,ݔ ሻൌܻ ሺݔሻ ݂ሺݐሻ. ሺ2.20ሻ Substituting Eqn. 2.20 in Eqn. 2.19 gives ߩ ܪܻሺܣݔᇱሺݔሻ ሺݔሻሻ ݂ሷሺݐሻെܣܻᇱሺݔሻ݂ሺݐሻെܻᇱᇱሺݔሻ݂ሺݐሻൌ 0. ሺ2.21ሻ Dividing Eqn. 2.21 by ܻ ሺݔሻ ݐ݂ሻ gives ߩሺ ܪ݂ ሺݐሻ݂ሷሺݐሻെሺݔሻܣᇱሺݔሻ ܣܻᇱሺݔሻ ሺݔሻܻെܻᇱᇱሺݔሻ ܻሺݔሻൌ 0. ሺ2.22 ሻ We can write, ݂ߩሷሺݐሻሺ݂ ܪሺݐሻ ሻ ⁄ ߣ ,so the time equation for Eqn. 2.22 becomes ݂ ሷሺݐሻ൅ܪ ߣ ߩൌെ ݂ሺݐሻൌ 0. ሺ2.23ሻ where ⁄ߩܪߣ ൌ߱ଶ. The spatial equation for Eqn. 2.22 can be written as ܻᇱᇱሺݔሻ൅ܣᇱሺݔሻ ܣሺݔሻ ܻᇱሺݔሻ൅ߩ ܪ߱ ଶ ܻሺݔሻൌ 0. ሺ2.24ሻ The wave number ߚ is defined as ߚଶൌ߱ଶ ߩ ܪ . ሺ2.25ሻ Substituting Eqn. 2.25 in Eqn.2.24 gives Texas ܻᇱᇱሺݔሻ൅ܣᇱሺݔሻ ܣሺݔሻTech University, Vijaya Kaithi, December 2008. 14 ܻᇱሺݔሻ൅ߚଶ ܻሺݔሻൌ 0. ሺ2.26ሻ Substituting Eqn. 2.20 in the boundary conditi ons in Eqn. 2.18, boundary condition at the surface of the Earth will be ܻሺ0ሻൌ0 and the boundary condition at the end of the elevator will be ܯ ௣ܻ ሷሺݐሻ൅ܣ ܪ ሺܮሻ ܻᇱሺܮሻ ݂ሺݐሻൌ 0. ሺ2.27ሻ ሺܮሻ ݂ Dividing Eqn. 2.27 by ݂ ሺݐሻ gives െܯ ௣ ߱ଶ ܻሺܮሻ൅ܣ ܪ ሺܮሻ ܻᇱሺܮሻൌ 0. ሺ2.28ሻ Above Eqn. can be written in term ܻᇱሺܮሻെ൬ܯ௣ ܣ ߩሺܮሻs of wave number as, ൰ሺߚଶሻܻሺܮሻൌ 0 ሺ2.29ሻ Cross-sectional area of th e elevator is given by ܣ ሺݎሻൌܣ ௅ ݁൭൬௥బమ ௛൰ቀଵ ௅ା௥ బ ି ଵ ௥ቁା൬௥బమ ଶ௛௥ ೞయ൰൫ሺ௅ା௥ బሻమି௥మ൯൱ . ሺ2.30ሻ where ܣ௅ is the cross-sectional area at the tip of the elevator. Substituting Eqn. 2.30 in the point m ss equation derived in Eqn, 2.16 a ܯ௣ൌܣ ߩ ௅ቌ݁ ׬൭൬௥బమ ௛൰ቀଵ ௅ା௥ బ ି ଵ ௥ቁା൬௥బమ ଶ௛௥ ೞయ൰൫ሺ௅ା௥ బሻమି௥మ൯൱ ݎ݃଴ଶ൬1 ݎଶെݎ ݎ௦ଷ ൰௥೑ ௥బݎ݀ቍ െݎ݃଴ଶቆ1 ݎ௙ଶെݎ௦ଷݎ௙ ቇ. ሺ2.31ሻ From Eqn. 2.31, we can observe that ܯ௣ܣ ߩሺܮሻ ⁄ is a function of length of the elevator ܮ and specific strength ݄ .Since ܣᇱሺݔሻܣሺݔሻ ⁄ is function of ݄ ,the spatial equation of the elevator in Eqn. 2. 26 is also a function of ݄ ݀݊ܽ ܮ . Therefore, the wave number for equations of motion of the elev ator is also a function of ݄ ݀݊ܽ ܮ only. It is difficult to solve the equations of motion in Eqn. 2.26 analytically due to varying cross-sectional area. Therefore, a num erical solution via fi nite difference method is used. The details of this method are give n in Appendix B. We let the forcing terms in Eqn. B.9 be zero for natural frequency calcula tions. The spatial equation in Eqn. 2.26 is Texas Tech University, Vi turned into an eigen value problem ሺሾܭሿെߣ ሾ ܯ ሿ ሻ ሼܻሽൌ ሼ0ሽ. The natural frequencies of the system are square root of eigen values, ߣ .jaya Kaithi, December 2008. 15 Figure 2.4 shows the variation of wave numbe r with the length of the elevator for different values of ݄ . To find the natural frequencies for l ongitudinal motion of the elevator, the ܪ term in ሾܭሿand ሾܯሿ matrices defined in Appendix B is replaced by Young’s modulus ܧ .In Figure 2.5, the first three natural periods of longitudinal vibration as functions of the elevator length for different values of ݄ are plotted. From this figure we can make the following observations. ¾ The first mode longitudinal natural peri ods for most of the elevator lengths will be around 6 hr . Natural periods for second and third modes of vibration are close to each other and under 2 hr. ¾ The natural periods are incr easing with the increase in ݄ ,but the rate of increase is very small. Similarly, the natural periods for transverse vibrations are also calculated using the [K] and [M] matrices from Eqn.B.9 in Appendix B. In this case, the ܪ term is replaced by the tensile strength ߪ .The natural periods of the transverse model are simply ඥ⁄ߪܧ times that of the longitudinal model. In Figure 2.6, the first three natural periods of transverse vibration are plotted as f unctions of elevator length for different values of ݄ .Following observations can be made from Figure 2.6. ¾ The natural periods of first mode of vibration for most of the elevator lengths are around 20 hr. Natural periods for second and third modes of vibration are around 5 hr. ¾ In contrast to the longitudinal case, natural periods are decreasing with the increase in ݄ for the transverse vibration. Texas Tech University, Vijaya Kaithi, December 2008. 16 Table 2.1: Taper ratio for different materials Material Tensile strength σ(GPa) Density ߩሺ݉݃݇ ଷ⁄ሻ Specific strength h(m) Taper ratio Carbon steel 0.25 7850 3250 ݁ଵହଶଷ Tool steel 2 7850 26 ൈ 1ଷ 0 ݁ଵଽ଴ Carbon nanotube [4, 5] 63 1300 5.05 ൈ 10଺ 2.72 Carbon nanotube [4, 5] 130 1300 10.2 ൈ 10଺ 1.62 Carbon nanotube [13] 300 1300 23.5 ൈ 10଺ 1.23 Texas Tech University, Vijaya Kaithi, December 2008. 17 Figure 2.1: Forces acting on an incremental element of length ݎ݀ Texas Tech University, Vijaya Kaithi, December 2008. 18 Figure 2.2: Cross-sectional area at various positions of the elevator for different values of ݄ Texas Tech University, Vijaya Kaithi, December 2008. 19 Figure 2.3: Relation between poi nt mass and elevator length for different values of ݄ Texas Tech University, Vijaya Kaithi, December 2008. 20 Figure 2.4: Variation of ߚ with length of the elevator Texas Tech University, Vijaya Kaithi, December 2008. 21 Figure 2.5: Natural periods of longitudinal vibration with length of the elevator Texas Tech University, Vijaya Kaithi, December 2008. 22 Figure 2.6: Natural periods of transverse vibration with length of the elevator Texas Tech University, Vijaya Kaithi, December 2008. 23 CHAPTER 3 NON-LINEAR MATHEMATICAL MODEL OF SPACE ELEVATOR 3.1 Introduction In this chapter, the space elevator is modeled as a long elastic cable in three dimensions. The three displacements in the axial ሺݑሻ, meridional ሺݒሻ and equatorial directions ሺݓሻ of the elevator are functions of both the Lagrangian co-ordinate ሺܺሻ along the elevator and time ሺݐሻ. The meridional direction po ints towards the North and equatorial direction points towards the West. The kinetic energy and potential energies of the elevator are obtained to form the Lagr angian. The three equations of motion are derived using Hamilton’s principle. Th ese three coupled and non-linear partial differential equations describe three-dimensi onal motion of the cable . Static solution of this model in the radial direction is considered and the cross-sectional area ܣሺܺሻ and counter mass ( ܯ௣ሻ are derived from that solution. 3.2 Equations of Motion In this section, we will derive a set of non-linear equations of motion for a cable that is similar to that ones used by Perkins and Mote [14] and Han and Benaroya [15]. Let ࢘ be the position vector from the center of the Earth to an incremental element and is given by ࢘ ൌ ሺݎ଴൅ܺ൅ݑ ሻ࢏ ൅ ࢐ݒ ൅ ,࢑ݓሺ3.1ሻ Where ݎ଴ = Radius of the Earth, Lagrangian coordinate, ܺ= ݖݕݔ co-ordinate system is at tached to the Earth and ࢑,࢐,࢏ are the unit vectors in the directions of ,ݔ and ݖ respectively, , ݕ ݑ is the displacement in the ݔdirection and is called the radial displacement , Texas Tech University, Vijaya Kaithi, December 2008. 24 ݒ is the displacement in the ݕ direction and is called the meridional splacement, when ߙ = 0 in Figure 3.1, di ݓ is the displacement in the ݖ direction and is ca lled the equatorial displacement., when ߙ = 0 in Figure 3.1. Figure 3.1 shows the ݖݕݔ co-ordinate system and Figure 3.2 shows displacements along the three directions. In this case the elevator is at an angle of ߙ with respect to the plane of the Earth’s equator. The angular velocity of the Ea rth about its rotational axis is ߗ ,so the co-ordinate system rotates with the angular velocity ߗ and vector form of the angular velocity is given by ࢹ ൌ ࢏ ߙ݊݅ݏߗ ൅ .࢐ ߙݏ݋ܿߗ ሺ3.2ሻ The kinetic energy of the evat is en by el or giv ܧܭ ൌ1 2ܯ௣࢖߲࢘ ݐ߲࢖߲࢘. ݐ߲൅ න12 ݉ሺܺሻ ߲࢘ ݐ߲߲࢘. ݐ߲ܺ݀ ௅ ଴, ሺ3.3ሻ where of the counterweight, ܯ௣ is the mass ሺܺሻൌ ܣ ߩሺܺሻ is the cable mass per unit le t ݉ ng h, ࢖ is the position vector of the point mass = ሺݎ଴൅ܮ൅ݑ ሻ࢏൅࢐ݒ ൅,࢑ݓ and ࢘ ܮ is the undeformed length of the elevator. The first term in Eqn 3.3 is kinetic energy due to the point mass and second term is kinetic energy due to the cable mass. The velocity vector, ⁄ݐ߲߲࢘ , is given by ߲࢘ ݐ߲ൌݑሶ࢏൅ݒ ሶ࢐൅ݓ ሶ࢑൅ ሺ ࢹ ൈ ࢘ ሻ ሺ3.4ሻ ൌ ሺݑ ሶ൅ߙݏ݋ܿ ߗݓ ሻ ࢏ ൅ ሺݒሶെ ߙ݊݅ݏ ߗݓ ሻ ࢐ ൅ሺݓ ሶ൅ ߙ݊݅ݏ ߗݒ െ ሺݎ଴൅ܺ൅ݑ ሻߙݏ݋ܿ ߗ ሻ ,࢑ and ⁄ݐ߲࢖߲࢘ ൌ ⁄ݐ߲߲࢘ |௅,௧ . Texas Tech University, Vijaya Kaithi, December 2008. 25 If the elevator is placed at the equator i.e. when α=0, the velocity vect or will be reduced to ߲࢘ ݐ߲ൌሺݑ ሶ൅ ߗݓ ሻ ࢏ ൅ ݒ ሶ࢐൅ሺݓ ሶ െ ሺݎ଴൅ܺ൅ݑ ሻ ߗሻ .࢑ሺ3.5ሻ The potential energy of the elevat or can be attributed from gr avitational and elastic forces acting on the elevator. The potential ener gy due to the elastic force is given by ܧܲ ௘ൌන ቌන෍෍ߪ ௜௝ ߝ݀ ௜௝ଷ ௝ୀଵଷ ௜ୀଵቍ ܸ݀ ௏. ሺ3.6ሻ Where ߝ௜௝ is the Green’s strain given by ߝ ൌ ݑ′൅1 2ቀݑ′ଶ൅ݒ′ଶ൅ݓ′ଶቁ, ሺ3.7ሻ and ߪ ௜௝ is the second Piola-Kirchoff stress. The first integral in Eqn.3.6 refers to th e integral over volume. Since the elastic potential energy acts in the ݔ direction , the integrand of the second integral ߪ௜௝ ߝ݀ ௜௝ can be reduced to ߪଵଵ ߝ݀ ଵଵ. From Hook’s law ߪଵଵൌߝ ܧ ଵଵ, so the elastic potential energy equation can be written as ܧܲ ௘ൌන12ߝ ܧଵଵଶܸ݀ ௏. ሺ3. 8ሻ The volume integral in Eqn 3.8 can be c onverted to line integral by writing the differential volume element as ܸ݀ ൌ ܣ ሺܺሻ ܺ݀ .Now, Eqn 3.8 can be expressed as ܧܲ ௘ൌන12ܣ ܧሺܺሻ ߝଵଵଶ ,ܺ݀௅ ଴ ሺ3.9ሻ Texas Tech University, Vijaya Kaithi, December 2008. 26 The gravitational potential energy can be at tributed from cable mass and point mass and it can be expressed as ܧܲ ௚ൌන െ ܣ ߩ ሺܺሻܯܩ ௘ ݎ ܺ݀௅ ଴െܯܩ ௘ܯ௣ ݎ௣ , ሺ3.10ሻ where ܯ௘ = Mass of the Earth, = magnitude of the position vector ࢘ = ඥሺݎ଴൅ܺ൅ݑ ሻଶ൅ݒଶ൅ݓଶ ݎ , and ݎ௣ = magnitude of the position vector to the point mass ห࢖࢘ หൌඥሺݎ଴൅ܮ൅ݑ ሻଶ൅ݒଶ൅ݓଶ . The first and second terms in the above equation indicates the gravitational potential energy due to cable mass and the counter we ight respectively. Total potential energy (ܧܲ )is ܧܲ ൌ ܧܲ ௘൅ܧܲ ௚. ሺ3.11ሻ Lagrangian will be given by ܮൌܧܭ െܧܲ ൌන ൬1 2 ܣߩሺܺሻ߲࢘ ݐ߲߲࢘. ݐ߲െ12ߝܣܧଶ൅ ܣߩሺܺሻܯܩ ௘ ݎ൰ ܺ݀௅ ଴ ൅12ܯ௣࢖߲࢘ ݐ߲࢖߲࢘. ݐ߲ ൅ܯܩ ௘ܯ௣ ݎ௣. ሺ3.12ሻ Hamilton’s principle states that, the Hamilt on’s Integral (I), th e variation of the Lagrangian L in equation 3.12 integrated over time should be equal to zero or ܫ ൌ ߜ න ܮ௧మ ௧భ ݐ݀ ൌ 0. ሺ3.13ሻ Substituting the Lagrangian L from equation 3.12 in the Hamilton’s Integral and simplifying gives Texas Tech University, Vijaya Kaithi, December 2008. 27 ܫൌߜනܮ௧మ ௧భ ݐ݀ ሺ3.14 ሻ ൌන න ቊ ቈܣߩሺܺሻሺെݑሷെ2 Ω ݓ ሶ ൅ሺݎ଴൅ܺ൅ݑ ሻߗଶሻ൅൫ ܣܧ ሺܺሻߝሺ1൅ݑᇱሻ൯ᇱ௅ ଴௧మ ௧భ െܣߩ ሺܺሻݎ݃଴ଶሺݎ଴൅ܺ൅ݑ ሻ ݎଷ൨ݑߜ ൅ ቈെܣߩ ሺܺሻݒሷ ൅ ሺሾܣܧሺܺሻݒߝᇱሿሻᇱെܣߩ ሺܺሻݎ݃଴ଶݒ ݎଷ቉ݒߜ ൅ቈܣߩሺܺሻሺെݓሷ൅2 ݑ ሶߗ൅ߗଶݓሻ൅ሺܣܧሺܺሻݓߝᇱሻᇱെܣߩ ሺܺሻݎ݃଴ଶݓ ݎଷ቉ݓߜቋݐ݀ ܺ݀ ൅න ەۖۖ۔ۖۖۓቈെܯ ௣ቆݑሷ൅2 Ω ݓ ሶ െሺݎ଴൅ܮ൅ݑ ሻߗଶ൅ݎ݃଴ଶሺݎ଴൅ܮ൅ݑ ሻ ݎ௣ଷቇ቉ݑߜሺݐ,ܮ ሻെ ቈܯ௣ቆݒሷ ൅ݎ݃଴ଶݒ ݎ௣ଷቇ቉ݒߜሺݐ,ܮ ሻെቈܯ௣ቆݓሷെ2 Ω ݑ ሶെߗଶݓെݎ݃଴ଶݓ ݎ௣ଷቇ቉ݓߜሺݐ,ܮ ሻ െሾܣܧሺܺሻߝሺ1൅ݑᇱሻሿݑߜ ฬܮ 0െሾܣܧሺܺሻݒߝᇱሿݒߜ ฬܮ0 –ሾܣܧሺܺሻݓߝ ᇱሿݓߜ ฬܮ0 ۙۖۖۘۖۖۗ ௧మ ௧భ . ݐ݀ From equation 3.14, we can deduce the thre e equations of motion and the boundary c s e m. The equations of motion are given by ondition for the spac elevator syste ܣߩሺܺሻሺݑ ሷ൅2 Ω ݓ ሶ െሺݎ଴൅ܺ൅ݑ ሻߗଶሻ ൌ൫ ܣܧ ሺܺሻߝሺ1൅ݑᇱሻ൯ᇱെܣߩ ሺܺሻݎ݃଴ଶሺݎ଴൅ܺ൅ݑ ሻ ݎଷ ሺ3.15ሻ ܣߩ ሺܺሻݒሷ ൌ ሺܣܧሺܺሻݒߝԢሻᇱെܣߩ ሺܺሻݎ݃଴ଶݒ ݎଷ ሺ3.16ሻ ܣߩ ሺܺሻሺݓ ሷെ2 ݑ ሶߗെߗଶݓሻൌሺܣܧሺܺሻݓߝԢሻᇱെܣߩ ሺܺሻݎ݃଴ଶݓ ݎଷ ሺ3.17ሻ and the boundary conditions are given by ݑ ሺ0, ݐሻൌ 0, ݒ ሺ0, ݐሻൌ 0, ݓ ሺ0, ݐሻൌ 0 ሺ3.18ሻ Texas Tech University, Vijaya Kaithi, December 2008. 28 ቈܯ௣ሺݑ ሷ൅2 Ω ݓ ሶ െሺݎ଴൅ܮ൅ݑ ሻߗଶሻ൅ܣܧ ሺܮሻߝሺ1൅ݑᇱሻ൅ܯ ௣ݎ݃଴ଶሺݎ଴൅ܮ൅ݑ ሻ ݎ௣ଷ቉ቤ ௅,௧ ൌ 0 ሺ3.19ሻ ቈܯ௣ݒሷ൅ܣܧ ሺܮሻݒߝԢ ൅ ܯ ௣ݎ݃଴ଶݒ ݎ௣ଷ቉ቤ ௅,௧ൌ 0 ሺ3.20ሻ ܯ ௣ሺݓ ሷെ2 Ω ݑ ሶെߗଶݓሻ൅ܣܧ ሺܺሻݓߝԢ ൅ ܯ ௣ݎ݃଴ଶݓ ݎଷ ௣ቤ ௅,௧ൌ 0. ሺ3.21ሻ The three equations of mo tion are coupled through ,ݎ,ߝ centripetal acceleration, gravity and the Coriolis terms. Therefore, it is not possible to have motion in one direction without the motion in the other directions. 3.3 Static Solution in the Radial Direction Let us find the static solution of Eqn 3.19 and derive the equations for the cross- sectional area and counter mass. From these equations, we will discuss the significance of the elongation factor ‘’ܥ . 3.3.1 Cross-sectional Area Let us derive the cross sectional area of elevator based on the static configuration of elevator while taking the elongation factor into consid eration. The elevator deforms from its original position because of the gravitational and centripetal forces acting on it. The co-ordinates of the elevator are defined from its base on the surface of the Earth. The initial co-ordinates of the elevator prior to its deformation are called Lagrangian co- ordinates, and are denoted as ܺ .The co-ordinates of elevator after the deformation are called Euler co-ordinates, and are denoted as ݔ .Elongation factor ( ܥ )depends on the tensile strength ሺߪሻ and Young’s modulus ( ܧ ,)and is given by Texas Tech University, Vijaya Kaithi, December 2008. 29 ܥ ൌ ඨ1൅2ߪ ܧ ൌ √1൅2 ߝ . ሺ3.22ሻ To illustrate the difference between ܺ and ݔ , let’s consider a portion of the space elevator as shown in figure 3.3. ܺ is the initial position of the elevator and ݔ is the position of the elevator af e mation. The strain is given by ter th de for ߝ ൌ1 2ݔ݀ଶെܺ݀ ଶ ܺ݀ଶ ݔ݀֜ ܺ݀ൌ√1൅2 ߝ . ሺ3.23ሻ Form Eqns. 3.22 and 3.23, we can write that, ݔ݀ ൌ ݔ֜ ܺ݀ ܥ ൌ ܺ ܥ Let us assume that the cross-sectiona l area of the elevator is a function of ܺ . The cross-sectional area is derived by assumi ng that the static stress at any point ܺis constant and equal to σ . The static transverse displacements, ܸ௘௤௨௜௟ ܹ ݀݊ܽ ௘௤௨௜௟ , are zeros and the radial static displacement, ܷ௘௤௨௜௟, is obtained by solving the following equation ߪ ൌ ߝܧ ௘௤௨௜ , ሺ3.24ሻ ௟ where ߝ௘௤௨௜௟ is given by ߝൌܷ Ԣ ௘௤௨௜௟ ൅ଵ ଶܷԢ௘௤௨௜௟ଶ. ሺ3.25ሻ Substituting Eqn 3.25 into Eqn 3.24 gives ߪ ൌ ܧ ൬ܷᇱ ௘௤௨௜௟ ൅1 2ܷᇱ ௘௤௨௜௟ଶ൰. ሺ3.26 ሻ The above equation can be written as ܷᇱ ௘௤௨௜௟ଶ ൅2 ܷᇱ ௘௤௨௜௟ െ2ߪ ܧൌ 0, ሺ3.27 ሻ and it can be solved with the initial condition ܷ௘௤௨௜௟ ሺܺൌ0 ሻൌ0. The solution gives the static radial displacement and can be written as, Texas Tech University, Vijaya Kaithi, December 2008. ܷ ௘௤௨௜௟ ൌሺܥെ1 ሻ ܺ ሺ3.28 ሻ 30 The radial displacement at the free end is (C-1)L , this can be quite significant because L is a very large number. If the elevator is in equilibrium and static stress is constant and equal to σ, Eqn 3.15 is reduced to െܣߩ ሺܺሻ ሺݎ଴൅ܺܥ ሻߗଶൌܣߪܥ ᇱሺܺሻ െ ܣߩ ሺܺሻݎ݃଴ଶ ሺݎ଴൅ܺܥ ሻଶ ሺ3.29 ሻ From the above equation, the ratio ܣᇱሺܺሻ⁄ܣሺ ሻ can be written as, ܺ ܣ ᇱሺܺሻ ܣሺܺሻൌݎ଴ଶ ܥ݄ቆ1 ൅ܺܥ ሻଶ ሺݎ଴െሺݎ଴൅ܺܥ ሻ ݎ௦ଷቇ. ሺ3.30 ሻ Integrating the above equation from ௦ ܺ ݋ݐ , ܺ ሾlog ௘ܣሺܺሻሿ|௑೛௑ൌቈെ ݎ଴ଶ ܥ݄ଶቆ1 ሺݎ଴൅ܺܥ ሻ൅ሺݎ଴൅ܺܥ ሻଶ 2ݎ௦ଷቇ቉ቤ ௑೛௑ , ሺ3.31 ሻ where ܺ௣ is the Lagrangian co-ord inate of the incremental el ement at any point along the elevator. The two convenient places are the GEO, ܺ௣ൌܺ ௦, and the surface of the Earth, ܺ௣ൌ0. Eqn 3.31 can be solved for in terms of the cross-sectional area of the elevator as ܣ ሺܺሻൌܣ ௦௘ቌష ೝబమ ೓಴మቆభ ሺೝబశ಴೉ ሻ శ ሺೝబశ಴೉ ሻమ మೝೞయቇቍ ௘ቌష ೝబమ ೓಴మቆభ ሺೝబశ಴೉ ೞሻ శ ሺೝబశ಴೉ ೞሻమ మೝೞయቇቍ or A0e൮‐ r02 hC2ቌ1 ൫r0൅CX ൯ ൅ ൫r0൅CX ൯2 2rs3ቍ൲ eቌ‐ r02 hC2ቆ1 r0 ൅ r02 2rs3ቇቍ. ሺ3.32ሻ The Eqn 3.32 is plotted in Figure 3.3 as a function of the current distance from Earth’s surface, x, where the Eulerian co-ordinate is re lated to the Lagrangian coordinate by x = CX . The area is normalized by the area at th e Earth’s surface. This plot shows the variation in the cross-sectional area of the elevator, when th e elongation factor is taken in Texas Tech University, Vijaya Kaithi, December 2008. 31 to account. From Figure 3.3, we can observe that the variation in the cross-sectional area due to elongation factor is quite significant. 3.3.2 Derivation of Countermass Let us consider an incremental element of mass ݉݀ as shown in Figure 3.5. The forc e nt are es acting on this el me given by, ՛ Σܨ ௥௔ௗ௜௔௟ ൌܶ݀ െܨ ௚ൌܽ ݉݀ ௥ ሺ3.33 ሻ Where ܶ݀ is the internal tension wh ich varies along the elevator, ܨ௚ is the gravitational pull given by ܨ௚ൌ ܯ ܩ ௘ ݎ/݉݀ଶ, and a r is the radial acceleration given by ܽ௥ ൌ െ ݒଶ/ݎ . Eqn. 3.33 can be rewritten as ܶ݀ ൌܯܩ ௘݉݀ ݎଶെݒ ݉݀ଶ ݎൌݎ ݃ ଴ଶܣ ߩሺܺሻ ܺ݀ ቆ1 ሺݎ଴൅ܺܥ ሻଶെሺݎ଴൅ܺܥ ሻ ݎ௦ଷቇ ሺ3.34ሻ For the elevator to be in static equilibriu m, the tension at the tip of the elevator must be zero or T(L)=0. For ܥ = 1, we recover the value ܮ = 144000 km . For ܥ =1.0583, the undeformed length of the elevator is ܮ =136000 km. The deformed length of the elevator is ݈ =ܮܥ =144000 km. Note that the deformed length of the elevator is always 144000 km for any value of ܥ . When the point mass is included, the length of the elevator can be shortened. The free body diagram of forces acting on th e point mass is shown in Figure 3.6. The summation of forces acting o point n the mass is given by ՛Σ ܨ ௥௔ௗ௜௔௟ ൌെ ܶ൅ܨ ௚ൌܯ ௣ ܽ௥, which results in ܯ ௣ݎ ݃଴ଶቆ1 ሺݎ଴൅ܮܥ ሻଶെሺݎ଴൅ܮܥ ሻ ݎ௦ଷቇ ൅ ܶሺܮሻ ൌ 0. ሺ3.35ሻ Texas Tech University, Vijaya Kaithi, December 2008. 32 From Eqn 3.35, point mass can be written as, ܯ ௣ൌܶ ݎ ݃଴ଶ൬ሺݎ଴൅ܮܥ ሻ ݎ௦ଷ െ1 ሺݎ଴൅ܮܥ ሻଶ൰ ሺ3.36ሻ The tension acting on the point mass can be found by integrating Eqn. 3.34 or ܶ ൌ න ܶ݀ ௅ ଴ ሺ3.37ሻ ൌ ۉۇܣߩ ௦ݎ݃଴ଶ ݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ೞሻ ା ሺ௥బା஼௑ ೞሻమ ଶ௥ೞయቇ൱ یۊන݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ሻାሺ௥బା஼௑ ሻమ ଶ௥ೞయቇ൱ ቆ1 ሺݎ଴൅ܺܥ ሻଶ௅ ଴ െሺݎ଴൅ܺܥ ሻ ݎ௦ଷቇ ܺ݀ Su tibs tuting Eqn 3.37 into Eqn 3.36 gives, ܯ௣ ሺ3.38ሻ ൌܣ௦ܥߪ ߗଶቆሺݎ଴൅ܮܥ ሻെݎ௦ଷ ሺݎ଴൅ܮܥ ሻଶቇ ۉۈۇ݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௅ ሻାሺ௥బା஼௅ ሻమ ଶ௥ೞయቇ൱ െ݁൭ି ௥బమ ௛஼మቆଵ ௥బା௥బమ ଶ௥ೞయቇ൱ ݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ೞሻ ା ሺ௥బା஼௑ ೞሻమ ଶ௥ೞయቇ൱ یۋۊ The point mass depends on As, C, L , and h. The dependency on As can be eliminated if the point mass is normalized by the total mass of the elevator. The total mass of the elevator is given by ݉ൌ නܣ ߩ ሺܺሻܺ݀௅ ଴ ൌܣߩ ௦ ݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ೞሻ ା ሺ௥బା஼௑ ೞሻమ ଶ௥ೞయቇ൱න݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ሻ ା ሺ௥బା஼௑ ሻమ ଶ௥ೞయቇ൱௅ ଴ ܺ݀ሺ3.39ሻ Texas Tech University, Vijaya Kaithi, December 2008. 33 Th ܯ௣ ݉e mass ratio is given by dividing Eqns. 3.38 and 3.39 or ൌܥ ݄ ݎ଴ଶ൬ሺݎ଴൅ܮܥ ሻ ݎ௦ଷ െ1 ሺݎ଴൅ܮܥ ሻଶ൰ ۉۈۇ݁൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௅ ሻାሺ௥బା஼௅ ሻమ ଶ௥ೞయቇ൱ െ݁൭ି ௥బమ ௛஼మቆଵ ௥బା௥బమ ଶ௥ೞయቇ൱ ݁׬൭ି ௥బమ ௛஼మቆଵ ሺ௥బା஼௑ ሻ ା ሺ௥బା஼௑ ሻమ ଶ௥ೞయቇ൱௅ܺ݀଴ یۋۊ .ሺ3.40ሻ Now, ܯ௣/݉ is a function of ,ܥ,݄ and ܮ .This equation is plotted in Figure 3.7 for two values of ܥ as a function of the elevator length, ܮ . For this figure, ݄is set to 4709 km, and the elongation factors are ܥ = 1.0583 and 1. From this figure, we observe that there is a significant deviation in the point mass to cable mass ratio, when elongation factor ሺܥሻ is 1.0583. Texas Tech University, Vijaya Kaithi, December 2008. 34 Figure 3.1: Co-ordinate syst em of the space elevator Texas Tech University, Vijaya Kaithi, December 2008. 35 Figure 3.2: Displacements in non-linear model of the space elevator [15] Texas Tech University, Vijaya Kaithi, December 2008. 36 Figure 3.3: Difference between Eule r and Lagrangian co-ordinates Texas Tech University, Vijaya Kaithi, December 2008. 37 Figure 3.4: Variation in the cr oss-sectional area of the elevator with elongation factor Texas Tech University, Vijaya Kaithi, December 2008. 38 Figure 3.5: Forces acting on an incremental element Figure 3.6: Forces acting on the countermass Texas Tech University, Vijaya Kaithi, December 2008. Figure 3.7: Variation in the point mass to cable mass ratio for different values of ܥ 39 Texas Tech University, Vijaya Kaithi, December 2008. 40 CHAPTER 4 TIDAL FORCES 4.1 Introduction In this chapter, we will discuss how the tidal forces are formed because of differential gravitational pull on different parts of the Earth. The co-ordinates of the Earth-Moon and Earth-Sun systems are defi ned. Co-ordinate transformation matrices have been developed to convert the Earth- Moon and the Earth-Sun Co-ordinate systems into a co-ordinate system that rotates with the Earth. Equatio ns of the tidal accelerations due to Moon and the Sun are derived, whic h are divided into three components acting along the axial, equatorial and meridional dire ctions. These tidal accelerations have been plotted with the time for hundred hours. Forc ing periods of the tidal accelerations have been found from their PSD plots. 4.2 Formation of Tidal Forces According to Newton’s law of gravita tion, any two bodies exert gravitational force of attraction on each other and the force will act along the line joining the two bodies. The magnitude of this force is proporti onal to the product of the masses of two bodies and inversely proportional to the square of the distance between them. Considering the Earth-Moon system, the Earth exerts a gravitational force on the Moon and the Moon also exerts the equal gravita tional force on the Earth. The equation for the gravitational force of attraction be e the Earth and the Moon is twe n ܨ ൌܯܩ ௘ܯ௠ ݎଶ, ሺ4.1ሻ where ܯ ௘ = Mass of the Earth, Texas Tech University, Vijaya Kaithi, December 2008. 41 ܯ = Mass of moon, ௠ = Newton’s gravitational constant =6.67× 10ିଵଵ ݉ ܰଶ݃݇ଶ⁄ , and ܩ ݎ = distance between center of the Earth to the cen ter of moon. The Earth exerts gravitational force on the Moon, which causes the moon to revolve around the Earth. In a similar way, the Moon also exerts gravitational force on the Earth causing it to revolve around the Moon. This combined effect causes the Earth and the Moon to revolve about a common point th at is their center of mass. This point is also called Barycenter [16]. From Table 4.1, we can observe that the Earth’s mass is almost 81 times greater than the mass of th e Moon. Since the Earth is massive than the Moon, barycenter lies close to the Earth. In fact, it is inside the Earth [16]. The Earth is so large that the effect of gravitational force due to the Moon will be different at different locations on the Earth. The part of the Earth closer to the Moon will experience greater acceleration due to the Moon’s gravitationa l pull than the one farther from the moon. This creates differential grav itational force across the Earth. As shown in Figure 4.1, point A experiences more gravitati onal pull from the Moon than point B. And point B experiences more gravitational pull than point C (The size of arrows indicates the magnitude of the acceleration; the figure is not up to scale). The shaded region in Figure 4.1 indicates the tidal bulge which occurs beca use of the difference in the gravitational pulls by the Moon at points A and C. These forces will try to deform the shape of the Earth along its equator to make it into an egg shaped object. But these forces ar e very small compared to the internal forces (Mechanical and gravitational forces) of the Earth, so the Earth deforms only about few centimeters [17]. But water is in liquid state, so it experiences the effect of tidal forces. These ocean waters start flowing towards two opposite ends of the Earth, one near the moon and the other one away from the moon as shown in Figure 4.2. These are called Tidal bulges. Since the Earth takes one day to make complete rotation about its own axis Texas Tech University, Vijaya Kaithi, December 2008. 42 and tidal bulges move with respect to the Moon, a particular place on the Earth experiences two high tides and two low tides every day. But by the time the Earth returns to its original position after 24 hours, th e Moon would have displaced by an angle 13.19 °. Earth must turn approximately 53 minutes more after reaching its initial position to get to the new position of the tidal bulge. This happe ns because moon is orbiting the Earth with a period of 27.3 days. In our solar system, the Sun is the next major source after the Moon that exerts considerable tidal gravitational force on the Earth. The Sun is massive than the Earth (Table 4.1), but the Moon is much closer to the Earth than the Sun (Table 4.2). According to Newton’s law of gravitation, gravitationa l force is proportional to mass but inversely proportional to cube of distance between two objects. The Sun’s share of the total tidal gravitational force acting on the Earth is only 30%. The tidal effect increases when the Earth, the Moon and the Sun come in a straight line. This happens during New Moon and the Full moon. The ocean tides produced during this period are called Spring tides. The tidal effect on the Earth decreases when the gravitational force vectors from the Sun and the Moon are perpendicular to each other. This occurs dur ing the first-quarter phase and third-quarter phases of the Moon. The tides produced dur ing this period are called Neap tides. Formation of the Spring tides and Neap tides is explained in Figure 4.3. 4.3 The Earth-Moon Co-ordinate System Let us first define coordinate systems as shown in Figure 4.4 in order to formulate the tidal forces due to the Moon. Let ݖݕݔ frame be attached to the Earth so that the ݖݕݔ frame rotates once every 24 hours, or with an angular velocity of ߗ ൌ 2/ߨሺ24 ൈ 60 ൈ 60ሻ ൌ 7.27 ൈ 10⁻⁵ሺݏ/݀ܽݎሻ . The unit vectors associated with the ݖݕݔ frame are ,࢐,࢏ and .࢑ The ݕ axis is aligned with the North pole, ݔ axis is normal to the Earth's surface Texas Tech University, Vijaya Kaithi, December 2008. 43 pointing directly at the incremental mass, a nd z axis is along the equator as shown in Figure 4.4. The position vector to the point mass can be written as ࢘ ൌ ࢏ݎ .We can back out to the inertial frame ܾܿܽ ,by rotating the ݖݕݔ frame about the ݕ axis by െݐߗ . The ܾaxis of the inertial frame ܾܿܽ is still aligned with Earth' s North Pole, and the plane of Earth's equator is on the ܿܽ plane. Note that the unit vectors associated with the ܾܿܽ frame are ,࢈,ࢇ and ࢉ .The relationship between unit vectors ࢉ,࢈,ࢇ and ࢑,࢐,࢏ is given by a matrix ሾܴଵሿ defined by ቈࢇ ࢈ ࢉ቉ൌ൥ ݏ݋ܿሺݐߗሻ 0 ݊݅ݏሺݐߗሻ 01 0 െ ݊݅ݏሺݐߗሻ 0 ݏ݋ܿሺߗ൩ ൥࢏ ࢐൩. ሺ4.2ሻ ݐሻ ࢑ The ܾܿܽ frame is rotated about the ܿ axis by 28.65° , so that the new inertial frame, ܥܤܣ , is now aligned with the orbital plane of the Moon about the Earth. Now, the Moon's orbital plane is on the ܥܣ plane, and the ܤ axis is perpendicular to the orbital plane. The unit vectors associated with the ܥܤܣ frame are ࡮,࡭ and ࡯ . The relationship between the unit vectors ࢉ࡮,࡭ is given by a ma ix ଶሿ defined by ,࡯ and ,࢈,ࢇ tr ሾܴ ൥࡭ ࡮ ࡯൩ൌ൥ ݏ݋ܿሺ28.65ሻ ݊݅ݏሺ28.65ሻ 0 െ ݊݅ݏሺ28.65ሻ ݏ݋ܿሺ28.65ሻ 0 00 1൩ ቈࢇ ࢈ ࢉ቉. ሺ4.3ሻ Finally, ܥܤܣ frame is rotated by ߱௘௠ݐ about the ܤ axis to obtain the ܼܻܺ frame that is attached to the Earth-Moon syst em. The unit vectors associated with the ܼܻܺ frame are ࡶ,ࡵ and ࡷ . The relationship between the unit vectors ࡷ,ࡶ,ࡵ and ࡯,࡮,࡭ is given by a matrix ሾܴଷሿ de ned bfi y ൥ࡵ ࡶ ࡷ൩ൌ൥ݏ݋ܿሺ߱௘௠ݐሻ0 െ ݊݅ݏ ሺ߱௘௠ݐሻ 01 0 ݊݅ݏሺ߱௘௠ݐሻ0ݏ݋ܿ ሺ߱௘௠ݐሻ൩ ൥࡭ ࡮ ࡯൩. ሺ4.4ሻ The relationship between different refere nce frames is shown in Figure 4.5 and relationship between the unit vectors ࡷ,ࡶ,ࡵ and ࢑,࢐,࢏ is given by, Texas Tech University, Vijaya Kaithi, December 2008. 44 ൥ࡵ ࡶ ࡷ൩ൌሾܴଷሿ ሾܴଶሿ ሾܴଵሿ ൥࢏ ࢐ ࢑൩ ሺ4.5ሻ 4.4 Equation of Tidal Forces due to the Moon The co-ordinate system of the Earth, sp ace elevator and the Moon is explained in Figure 4.6. Let us consider an incremental mass ݉݀ ,which is at a distance ݎ away from the center of the Earth and also on the plane of the Earth’s equator. The Earth and the Moon rotate about their own axes and the Moon revolves around the Earth. These rotations are included by considering that ¾ The Earth rotates about its ow n axis by angular velocity ߗ ¾ The rotation of the Earth about the Sun is neglected. ¾ The Earth and the Moon are orbi ting around their center of mass (barycenter) with angular velocity ߱௘௠ The center of mass o rth yst s f the Ea - Mo on s em is calculated as follow ܯ ௘ݏ௘௠ൌܯ ሺ െݏ ௘௠ሻ, ௠ܦ௘௠ ݏ ௘௠ൌܯ௠ܦ௘௠ ܯ௠൅ܯ ௘ , ሺ4.6ሻ where ܯ = Mass of the Moon, ௠ ௘ = Mass of the Earth, ܯ ܦ = Distance between centers of the Earth and the Moon, and ௘௠ ݏ ௘௠ൌ Distance between center of the Earth and center of mass of the Earth- Moon system. Texas Tech University, Vijaya Kaithi, December 2008. 45 The angular velocity with which the Moon orbits the Earth is obtained by equating gravitational pull with th e centripetal acceleration ܯܩ ௠ܯ௘ ܦ௘௠ଶൌ ܯ ௘ݏ௘௠߱௘௠ଶൌ ܯ ௘ሺܦ௘௠െݏ ௘௠ሻ߱௘௠ଶ, ߱ ௘௠ଶൌܯܩ ௠ ݏ௘௠ܦ௘௠ଶൌ ܩሺܯ ௠൅ ܯ ௘ሻ ܦ௘௠ଷ . ሺ4.7ሻ We have approximated the space elevat or with a point mass at a distance ݎ from center of the Earth. The forces acting on poi nt mass are due to the Moon and rotation about the center of mass of Earth-Moon rotating system. The equation for the tidal forces acting on elevator per unit mass due to the Moon is given by ܲ௠௢௢௡ ݉݀ൌ െ ݏ ௘௠߱௘௠ଶ ࡵ ൅ ܯܩ ௠ ܴ௘௠ଷ ሺܦ௘௠ࡵെ࢏ݎ ሻ, ሺ4.8ሻ Where ߩ Density of material used for space elevator, = = Cross –sectional area of the elevator, and ܣ ܲ ௠௢௢௡ = Tidal forces acting on the elevator due to the Moon. ܲ௠௢௢௡ ⁄݉݀ is called the differential acceleration felt by point mass due to the Moon. After replacing the I in Eqn. 4.8 with the expression in Eqn. 4.5, the differential acceleration equation becomes Texas Tech University, Vijaya Kaithi, December 2008. 46 ܲ௠௢௢௡ ݉݀ ൌ൥൭ቆܯܩ ௠ܦ௘௠ ܴ௘௠ଷ –ݏ ௘௠߱௘௠ଶቇ ሺcosሺ߱௘௠ݐሻcosሺݐߗሻcosሺ28.65 ሻ൅s i n ሺ߱௘௠ݐሻsinሺݐߗሻሻ൱ െቆ ܯܩ ௠ݎ ܴ௘௠ଷቇ൩ ࢏ ൅ ቈቆܯܩ ௠ܦ௘௠ ܴ௘௠ଷ –ݏ ௘௠߱௘௠ଶቇ ሺcosሺ߱௘௠ݐሻsinሺ28.65 ሻሻ቉࢐ ൅ቈቆܯܩ ௠ܦ௘௠ ܴ௘௠ଷ –ݏ ௘௠߱௘௠ଶቇ ሺcosሺ߱௘௠ݐሻsinሺݐߗሻcosሺ28.65 ሻ െs i n ሺ߱௘௠ݐሻcosሺݐߗሻሻ቉ .࢑ ሺ4.9ሻ ܲ௠௢௢௡ is divided into three components ܲ௨,ܲ௩,ܲ௪, which will act in the axial, equatorial and meridional directions respectively. So i, j, k components of the Eqn. 4.9 are the differential accelerations in axial, equatorial and meridional directions. Figure 4.7 shows the differential acceleratio ns felt by the elevator with time at different positions in different directions due to the Moon. We can clearly observe that as we move away from the Earth, diffe rential accelerations are increasing. ܲ௨,ܲ௪ curves are periodic with period of 12.5 hr. They are out of phase by angle 180°. The differential acceleration along cons tant longitude ሺܲ௩ሻ is periodic with a period of 25 hr. 4.5 Equation of Tidal Forces due to the Sun The Sun is the other main source for tid al forces apart from the Moon. The Sun contributes up to 30% of the total tidal forces acting on the Earth. The equation for the forces acting on the elevator due to the Sun can be derived in a similar way as we have done for the Moon. But the angle between the Earth’s equator and or bital plane of the Earth is 23.5 ° as shown in figure 4.4. So the differe ntial acceleration due to the Sun is given by Texas Tech University, Vijaya Kaithi, December 2008. 47 ܲ௦௨௡ ܣߩൌ െ ݏ ௘௦߱௘௦ଶ ࡵ ൅ ܯܩ ௦௨௡ ܴ௘௦ଷ ሺܦ௘௦ࡵെ࢏ݎ ሻ, ሺ4.10ሻ where ܲ ௦ = Forces acting on elevator due to the Sun, ௨௡ ௦௨௡ = Mass of the Sun, and ܯ ܦ ௘௦ = Distance between centers of the Earth and the Sun. The distance between center of the Eart h and center of mass of the Earth- Sun system ሺݏ௘௦ሻ is given by ݏ ௘௦ൌܯ௦ܦ௘௦ ܯ௦൅ܯ ௘ . ሺ4.11ሻ The angular velocity ሺ߱௘௦ሻ of the Earth –Sun system about their center mass is given by ߱ ௘௦ଶൌܯܩ ௦ ݏ௘௦ܦ௘௦ଶൌ ܩሺܯ ௦൅ ܯ ௘ሻ ܦ௘௦ଷ. ሺ4.12ሻ For Earth-S IJK g en by un system, the relation between reference frames and ijk is iv ൥ࡵ ࡶ ࡷ൩ൌ൥cosሺ߱௘௦ݐሻ0െ s i n ሺ߱௘௦ݐሻ 01 0 s ݐሻ൩൥cosሺ23.5 ሻ sinሺ23.5 ሻ0 െs i n ሺ23.5 ሻcosሺ23.5 ሻ0 00 1൩ inሺ߱௘௦ݐሻ0c o s ሺ߱௘௦ ൥ ݏ݋ܿሺݐߗሻ 0 ݊݅ݏሺݐߗሻ 01 0 െ ݊݅ݏሺݐߗሻ 0 ݏ݋ܿሺݐߗሻ൩ ൥࢏ ࢐ ࢑൩ ሺ4.1 3ሻ After replacing the I in Eqn 4.10 with the expression in Eqn. 4.13, the differential acceleration equation becomes Texas Tech University, Vijaya Kaithi, December 2008. 48 ܲ௦௨௡ ܣߩൌ൥൭ቆܯܩ ௦ܦ௘௦ ܴ௘௦ଷ –ݏ ௘௦߱௘௦ଶቇ ሺcosሺ߱௘௦ݐሻcosሺݐߗሻcosሺ23.5 ሻ൅s i n ሺ߱௘௦ݐሻsinሺݐߗሻሻ൱ െቆ ܯܩ ௦ݎ ܴ௘௦ଷቇ൩ ࢏ ൅ ቈቆܯܩ ௦ܦ௘௦ ܴ௘௦ଷ –ݏ ௘௦߱௘௦ଶቇ ሺcosሺ߱௘௦ݐሻsinሺ23.5 ሻሻ቉࢐ ൅ቈቆܯܩ ௦ܦ௘௦ ܴ௘௦ଷ –ݏ ௘௦߱௘௦ଶቇ ሺcosሺ߱௘௦ݐሻsinሺݐߗሻcosሺ23.5 ሻ െs i n ሺ߱௘௦ݐሻcosሺݐߗሻሻ቉ . ࢑ ሺ4.14ሻ ܲ௦௨௡ is divided into three components ܲ௨,ܲ௩,ܲ௪, which will act in the axial, equatorial and meridional directions respectively. So i, j, k components of the Eqn. 4.11 are the differential accelerations in axia l, equatorial and meridional directions. Figure 4.8 shows the differential acceleratio ns felt by the elevator with time at different positions in different directions due to the Sun. We can clearly observe that as we move away from the Earth, diffe rential accelerations are increasing. ܲ௨,ܲ௪ curves are periodic with period of 12 hr . They are out of phase by angle 180°. The differential acceleration along cons tant longitude ሺܲ௩ሻ is periodic with a period of 24 hr. The magnitudes of the differential accelerations cau sed by the Sun are less than half of those by the Moon. 4.6 PSD Plots of the Tidal Forces Power spectral density (PSD) is used to describe the distribution of power contained in a signal over frequency [18]. PSD of a stationary random process ݔ௡ is mathematically related to the correlation se quence by the discrete-time Fourier transform and is given by Texas Tech University, Vijaya Kaithi, December 2008. 49 ܲ ௫௫ሺ߱ሻൌ1 2ߨ ෍ܴ ௫௫ሺ݉ሻ ݁ି௝ఠ௠ஶ ௠ୀିஶ . ሺ4.15ሻ Since ൌ2 ݂݂ߨ ௦⁄ , the above equation can be expr essed as a function of physical frequency ݂ as ܲ ௫௫ሺ߱ሻൌ1 ݂௦ ෍ܴ ௫௫ሺ݉ሻ ݁ିଶగ௝௙௠ ௙ ೞ⁄ஶ ௠ୀିஶ. ሺ4.16ሻ We are going to use the PSD plots to detect the forcing periods from the sinusoidal plot of differential accelerations due to the Moon and the Sun. Eqn. 4.9 consists of three sinusoidal functions cosሺ߱௘௠ݐሻ, cos ሺݐߗሻ, cos ሺ28.65 ሻ indicating that it will have multiple periods. But from Figure 4.7 where the diffe rential acceleration due to the Moon is plotted, we can observe a forcing period of 12.5 hr in longitudinal and meridional directions and 25 hr in the equatorial direction. Fr om the PSD plot of differential accelerations due to the Moon in Figure 4.9, we can see that the differential accelerations have periods of 6.2 hr, 8.3 hr , 12.5 hr and 25 hr . Eqn. 4.14 which explains the differential accelera tions due to the Sun has three sinusoidal functions cosሺ߱௘௦ݐሻ, cos ሺݐߗሻ, cosሺ23.5 ሻ. But from Figure 4.9 we cannot identify all the forcing periods. From the PSD plot of differe ntial accelerations due to the Sun in Figure 4.9, we can see that the differentia l accelerations have periods of 6 hr, 8 hr , 12.5 hr and 25 hr . Texas Tech University, Vijaya Kaithi, December 2008. 50 Table 4.1: Mass of the Moon, the Earth & the Sun Mass of the Moon Mass of t rth he Ea Mass of the Sun 0.07349 × 10ଶସ 5.9742 × 10ଶସ 1.989 × 10ଷ଴ Table 4.2: Distance from the Earth to the Moon & the Sun Distance between the Earth and the Moon Distance between the Earth and the Sun 3.84467 × 10଼ ݉ 1.496 × 10ଵଵ Texas Tech University, Vijaya Kaithi, December 2008. 51 Figure 4.1: Gravitational pull on the Eart h by the Moon causing tidal bulges [19] Figure 4.2: Displacement in the positi on of the Moon after 24 Hrs [16] Texas Tech University, Vijaya Kaithi, December 2008. Figure 4.3: Spring and Neap tides [17] Figure 4.4: The Earth- Moon- Sun system 52 Texas Tech University, Vijaya Kaithi, December 2008. 53 Figure 4.5: Relation between di fferent reference frames Texas Tech University, Vijaya Kaithi, December 2008. 54 Figure 4.6: The Earth- Moon co-ordinate system including the space elevator Texas Tech University, Vijaya Kaithi, December 2008. 55 Figure 4.7: Differential accelerations at differe nt positions of elevator due to the Moon Texas Tech University, Vijaya Kaithi, December 2008. 56 Figure 4.8: Differential accelerations at differe nt positions of elevator due to the Sun Texas Tech University, Vijaya Kaithi, December 2008. Figure 4.9: PSD plot of differen tial accelerations due to the Moon 57 Texas Tech University, Vijaya Kaithi, December 2008. Figure 4.10: PSD plot of differen tial accelerations due to the Sun 58 Texas Tech University, Vijaya Kaithi, December 2008. 59 CHAPTER 5 RESULTS AND DISCUSSION 5.1 Introduction In this chapter, the response of the space elevator under the influence of tidal forces is calculated using linear and non-lin ear models. The equations of motion are given in Chapter 2 and 3. The finite differen ce method is used to calculate response of the space elevator under the influence of tidal for ces. The detailed description of the method is given in Appendix B and D. For the numerical results, we will use ܧ = 1 TPa and ρ = 1300 kg/m3, where E is Young’s modulus and ρ is the density. We will also use σ = 60 GPa , which is within the experimentally measured tensile strength range for CNT [4, 5]. This va lue is significantly lower than its theoretical limit 300 GPa [4, 5]. This value results in ݄ ൌ 4.7 ൈ 10଺ ݉ . We will vary the length of the elevator for the numerical simulations. 5.2 Design of the Space Elevator In this section, we will consider some design criterion for different lengths of the elevator. A particular range of length of the elevators that satisfy the particular design criterion will be suggested at the end of th at criterion. In the follo wing sections, we will develop some design criterions and come up with the opt imum elevator design. 5.2.1 Maximum Amplitude Criterion Let us consider responses at the point ma ss predicted by the linear models as first approximations. Recall that in this model th e motion in the longitudinal and transverse directions are decoupled and th e Coriolis and the centripetal accelerations are neglected. Figure 5.1 and Figure 5.2 shows the longitudinal and transverse displacements of the tip of the elevator as a function of time for di fferent elevator length s. Figure 5.3 explains displacements of the elevator in three directions for the nonlinear coupled model of the Texas Tech University, Vijaya Kaithi, December 2008. 60 elevator. Note that the natural frequencies of the system vary with elevator lengths as shown in Figure 5.4 and Figure 5.5. In general, the response amplitude will grow as one of the natural frequencies of the system appr oach one of the forcing frequencies. When the natural frequency exactly matches the forc ing frequency we have resonance where the response grows linearly with time without a bound. However, in th is case, it is not easy to predict the amplitude increase because one of the natural frequency may be approaching one of the forcing frequency while another na tural frequency may be getting further away from some other forcing frequency. It is cumbersome to numerically simulate the response for every elevator length to obt ain the response amplitudes. The upper bound on the response amplitudes can be obtained analytically using modal analysis. The mathematical details of the modal analysis are given in Appendix C. The modal analysis method is used to find the maximum displacements for different elevator lengths. The maximum displacement is given by the sum of the absolute values of the amplitudes of the modal displacements. This can only be the upper bound because as we know that the amplitude of sin( t) + 2sin(2 t) is less than the sum of the amplitudes or 3. Figure 5.6 represents the maximum displacemen ts of elevator for different lengths of the elevator. Longitudinal displacements for 53,050 km and 56,750 km are very high as the natural periods are close to the forcing periods of 6.2 hr and 6 hr . Similarly, there is a sudden raise in transverse displacements for elevators of length 54,750 km and 59150 km , as their natural periods are clos e to the forcing periods of 25 hr and 24 hr . The second transverse natural period coincides with the tidal frequencies at 6 hr and 6.2 hr when the lengths of the elevator are 74,400 km and 78500 km , respectively. The resonance at these elevator lengths are shown in Figure 5.6 for transverse displacement. Note that the resonance at the higher mode has shar per peaks affecting smaller range of L. That is, it is easier to avoid the resonance at the higher mode by adjusting the elevator length slightly. It is interesting to note that both the longitudinal and the transverse amplitudes increase with increasing length. This ag rees with the trend shown in the response plots in Figure Texas Tech University, Vijaya Kaithi, December 2008. 61 5.1 and 5.2. Once we decide the acceptable response amplitudes, we can deduce the range of acceptable length of el evator for the design purposes. 5.2.2 Point Mass Criterion Figure 5.7 shows the point mass required for different lengths of the elevator so that the elevator will be in tension. The point mass is used to shor ten the length of the elevator. The point mass to the cable mass ratio decreases with the increase in the length of the elevator. ¾ The point mass required for lengt hs of the elevator from 35,786 km to 53,000 km is more than the mass of the cab le. During the construction of the elevator, sending a counter mass weighing more than the mass of the cable to an altitude of more than 35,000 km is not possible. Therefore, we are not going to consider elevator lengths of below 53,000 km in our further designs. ¾ Even though point mass required for elevators of length more than 100,000 km is very low, constructing elevators of such a height is not practically feasible. The rate of decr ease in the counter mass requi red for elevator lengths beyond 100,000 km is very small, so increasing the length of the elevator beyond that point is not recommended. From point mass criterion, we have elimin ated lengths of the elevator below 53,000 km and beyond 100,000 km from our design criterion. Therefore, we will consider lengths of the elevator between 53,000 km and 100,000 km in our next criterion. 5.2.3 Resonance The forcing periods of the tidal forc es due to the Moon and the Sun were calculated in Chapter 4. Tida l periods of the Moon are 6.2 hr, 8.3 hr, 12.5 hr and 25 hr . Tidal periods of the Sun are 6 hr, 8 hr , 12 hr and 24 hr. Texas Tech University, Vijaya Kaithi, December 2008. 62 ¾ The transverse natural periods of firs t, second and third modes of vibration of the elevator are plotted in Figure 5.5 for different lengths of the elevator between 53,000 km and 100,000 km . Resonance will occur when the natural periods of the elevator are equal or close to the forcing periods of the tidal forces. The transverse natural periods of first m ode of vibration for elevators of length between 53,000 km and 63,000 km are around 24 hr and 25 hr , which are the first forcing periods of tidal forces due to the Sun and the Moon respectively. ¾ The transverse natural periods of seco nd mode of vibration for elevators of length between 68,000 km and 85,000 km are close to the forcing periods of tidal forces due to the Sun (6 hr) and the Moon (6.2 hr). These regions are indicated by a rectangle in the figure. ¾ The longitudinal natural periods of firs t mode of vibration are close to the forcing periods of the tidal forces due to the Sun and the Moon for elevator lengths between 53,000 km and 60,000 km as shown in Figure 5.4 From above discussions, we can eliminat e lengths of the elevator between 53,000 km and 63,000 km and 68,000 km and 85,000 km from our design criterion. Therefore, we will consider lengths of the elevator between 63,000 km and 68,000 km and 85,000 km and 100,000 km in our next criterion. To illustrate the difference between linear and nonlinear models of the elevator, let us consider an elevator of length 82,000 km and plot the PSD of their responses. From the PSD plots of response in th e axial directions for linear and nonlinear models shown in Figure 5.8, we can observe that both of them have peaks in all the forcing frequency regions. From the nonlinear PSD plot, we can see that there is a peak at 4.63 hr indicating that it is the natural period of axial vibration. But the natu ral period for linear model is 5.3 hr , so there is a shift in the natural peri od of the nonlinear model from the linear model. Figure 5.9 and Figure 5.10 are linear and nonlinear PSD comparisons for the meridional and equatorial responses. From th ese plots we can observe that the natural Texas Tech University, Vijaya Kaithi, December 2008. 63 periods calculated from the linear model are consistent with those obtained from nonlinear model. 5.2.4 Dynamic Response of the Elevator In this section we will consider two differe nt design cases of the elevator such that they represent the different regi ons of lengths of the elevator mentioned in the last section and plot their responses and velo cities. The two design cases are 66,000 km and 91,000 km Case 1: Length of the Elevator is 66,000 km Figure 5.11 and 5.12 show the position and ve locity plots for the two-dimensional model of the elevator at the point mass. Th e maximum displacement of the point mass in the longitudinal direction is 120 m and maximum axial velocity of the point mass is 0.05 m/s. The maximum displacement of the point mass in the transverse direction is 6 km and the transverse velocity is 0.6 m/s. Figure 5.13 and 5.14 shows the position and velocity plots predicted by the t hree- dimensional model of the elevator respect ively. The maximum displacements of the elevator in axial, equatorial and meridional directions are 110 m, 10 km and 3 km , respectively. Maximum velocitie s of the elevator in axial, equatorial and meridional directions are 0.02 m/s, 0.5m/s and 0.04 m/s, respectively. When we compare the responses obtained using the two models, we find that both models predict similar amplitudes for the longitudinal motion. Howeve r, the transverse amplitude differs by a factor of 10/6. This shows that the two-dimensional linear models which do not take Coliolis acceleration, centripe tal acceleration, stretch in the elevator, and nonlinear coupling between the longitudinal and the tr ansverse motions are not adequate in predicting the response of the space elevator. In order to accurately predict the transverse motion, the three-dimensional nonlin ear modeling is inevitable. Texas Tech University, Vijaya Kaithi, December 2008. 64 Case 2: Length of the Elevator is 91,000 km Figure 5.15 and 5.16 are the response and ve locity plots for the two-dimensional model of the elevator. Maximum displacement of the elevator in the longitudinal direction is 230 m and maximum ve locity of the elevator is 0.05 m/s. The maximum displacement of the elevator in the transverse direction is 8 km and the velocity of the elevator is 0.6 m/s. Figure 5.17 and 5.18 shows the response and velocity plots for nonlinearly coupled three-dimensional model of th e elevator, respectively. The maximum displacements of the elevator in axial, equatorial and meridional directions are 230 m, 30 km and 6 km, respectively. The maximum velocities of the elevator in axial, equatorial and meridional directions are 0.05 m/s, 1.5m/s and 0.1 m/s, respectively. The transverse amplitude of 30 km may seem large. However, this is very small compared to the total length of the cable. In addition, the maximum ve locity that corresponds to this motion is small enough to permit necessary operations on the elevator. Again, we note that the transverse am plitudes, after taking into account the Coliolis acceleration, centripeta l acceleration, nonlinear coup ling between the transverse and the longitudinal motion, the stretch in the elevator, is about 30/8 times the one predicted by the simple linear model. Therefore, the linear model should not be used when more accurate dynamic results are required. 5.2.5 Accessible Destinations A space elevator is used to reach the destin ations within the Earth orbits as well as the solar orbits. Destinations within the Eart h orbits are easily reachable as they are thousands of kilometers away from the Ea rth. Space elevator acts as a sling to launch payloads to destinations which are trilli ons of kilometers away. Edwards [1] found a relation between the length of the elevator and the distance up to which it can launch the payloads. Figure 5.19 explains the relations hip between solar orbits accessible and Texas Tech University, Vijaya Kaithi, December 2008. 65 elevator length. As the length of the elevat or increases, the range of accessible solar orbits is increasing. From the discussion in Section 5.1.4, we found that the displacements and velocities of the elevator for design case 1 ar e half of those for cas e 2 and length of the elevator in case 2 is 27,000 km longer than the one in the case 1. From Figure 5.19 we can see that the elevator in case 1 can only be used for tran sportation of payloads within the Earth orbits where as the elevator in case 2 can launch payloads up to solar orbits. From Figure 5.19, we see that the elevator in case 2 (91,000 km long) can deliver payloads to Venus, Mars and Jupiter. Even though case 2 ha s higher displacements, its accessibility in space makes it a very useful desi gn. Therefore, depending on the type of destinations that should be reached either of the designs which satisfy that particular requirement should be selected. Texas Tech University, Vijaya Kaithi, December 2008. 66 Figure 5.1: Longitudinal displacements of th e pointmass for various lengths of the elevator for linear model Texas Tech University, Vijaya Kaithi, December 2008. 67 Figure 5.2: Transverse displacem ents of the poinmass for various lengths of the elevator for linear model Texas Tech University, Vijaya Kaithi, December 2008. 68 Figure 5.3: Displacements in the axial, meri dional, and equatorial directions of the pointmass for nonlinear model Texas Tech University, Vijaya Kaithi, December 2008. 69 Figure 5.4: Longitudinal natural periods of first, second and third modes of vibration predicted by the linear models for different lengths of the elevator indicating the regions of possible resonance Texas Tech University, Vijaya Kaithi, December 2008. 70 Figure 5.5: Transverse natural periods of first, second and third modes of vibration predicted by the linear models for different lengths of the elevator indicating the regions of possible resonance Texas Tech University, Vijaya Kaithi, December 2008. 71 Figure 5.6: Maximum displacements of the poi ntmass in the longitudinal and transverse directions for various le ngths of the elevator Texas Tech University, Vijaya Kaithi, December 2008. 72 Figure 5.7: Point mass over cable mass ratio for various lengths of the elevator Texas Tech University, Vijaya Kaithi, December 2008. 73 Figure 5.8: PSD plots for the longitudinal response of th e pointmass predicted by the linear and nonlinear models Texas Tech University, Vijaya Kaithi, December 2008. 74 Figure 5.9: PSD plots for the tr ansverse response of the pointmass predicted by the linear and nonlinear models Texas Tech University, Vijaya Kaithi, December 2008. 75 Figure 5.10: PSD plots for the transverse (equatorial) response of the pointmass predicted by the linear and nonlinear models Texas Tech University, Vijaya Kaithi, December 2008. 76 Figure 5.11: Displacements of the pointmass in the longitudinal and transverse directions for L = 66,000 km predicted by the linear model Texas Tech University, Vijaya Kaithi, December 2008. 77 Figure 5.12: Velocities of the pointmass in the longitudi nal and transverse directions for L = 66,000 km predicted by the linear model Texas Tech University, Vijaya Kaithi, December 2008. 78 Figure 5.13: Displacements of the pointmass in the axial, equatorial and meridional directions for L = 66,000 km predicted by the nonlinear model Texas Tech University, Vijaya Kaithi, December 2008. 79 Figure 5.14: Velocities of the pointmass in the axial, equatorial and meridional directions for L = 66,000 km predicted by the nonlinear model Texas Tech University, Vijaya Kaithi, December 2008. 80 Figure 5.15: Displacements of the pointmass in the longitudinal and transverse directions for L = 91,000 km predicted by the linear model Texas Tech University, Vijaya Kaithi, December 2008. 81 Figure 5.16: Velocities of the poi ntmass in the longitudinal and transverse directions for L = 91,000 km predicted by the linear model Texas Tech University, Vijaya Kaithi, December 2008. 82 Figure 5.17: Displacements of the pointmass in the axial, equatorial and meridional directions for L = 91,000 km predicted by the nonlinear model Texas Tech University, Vijaya Kaithi, December 2008. 83 Figure 5.18: Velocities of the pointmass in the axial, equatorial and meridional directions for L = 91,000 km predicted by the nonlinear model Texas Tech University, Vijaya Kaithi, December 2008. 84 Figure 5.19: Solar orbits accessible for di fferent lengths of the elevator [1] Texas Tech University, Vijaya Kaithi, December 2008. 85 CHAPTER 6 CONCLUSIONS AND FUTURE WORK In this thesis, we have introduced dynamic analysis into the design of the space elevator. After going through different de sign criterion, we have rounded upon two design cases. These two designs are se rving two different purposes. The first design case which has length of the elevator as 66,000 km will be primarily useful for delivering payloads up to thousa nds of kilometers away in space. The maximum displacement and velocity of the elevator in this case are 10 km and 0.6 m/s respectively. The point mass requi red at the tip of the elevator for this case is 0.47 times the mass of the cable. Sending this point mass to an altitude of 66,000 km will not be a problem during the deployme nt of the elevator. The second design case which has le ngth of the elevator as 91,000 km will be useful for delivering payloads within the Ea rth orbits as wells as to so lar orbits. This elevator can reach destinations in Venus, Mars and Jup iter. The maximum displacement and velocity of the elevator in this case are 30 km and 1.5 m/s respectively. The point mass required at the tip of the elevator for this case is 0.15 times the mass of the cable. Sending this point mass to an altitude of 91,000 km will not be a problem during the deployment of the elevator. Linear model results are accurate when examin ing the natural frequencies. For the actual response, the nonlinear coupled 3D modeling is necessary because the linear 2D model severely underestimates the amplitude of the transverse response. Resonance due to the parked elevator cars can be considered into the design of the space elevator as these cars can change the funda mental periods of the elevator. Transient response due to the accelerating elevator car can be investigated as this should be very small to ensure safe operation. Stability of the final design of the elevator in axial, equatorial and meridional dir ections can be checked. Texas Tech University, Vijaya Kaithi, December 2008. 86 REFERENCES [1] Edwards, B.C., The Space elevator, Phase I study , NASA Institute for Advanced Concepts, Report June, 2000. [2] Smitherman, Jr, D.V., Space Elevators: An Advanced Earth-Space Infrastructure for the New Millennium , NASA, Report August, 2000. [3] Iijima, S., “Helical micr otubules of graphitic carbon”, Nature, vol.354, p56, 1991. [4] Yu, M.-F., Lourie, O., Dyer, M.J., Moloni, K., Kelly, T.F. and Ruoff, R.S., “Strength and breaking mechanis m of multiwalled carbon nanotubes under tensile load,” Science, vol.287, pp 637-640, 2000. [5] Yu, M. F., Files B.S., Arepalli S. and Ruoff R.S., Physical Review Letters 84, 5552, 2000. [6] Artsutanov, Y., V Kosmas na Elektrovoze, Komsomolskaya Pravda, (contents described in Lvov). Science , 158, 946, 1967. [7] Isaacs, J., Vine, A. C., Bradner, H. and Bachus, G.E., “Satellite Elongation into a true Sky-Hook,” Science , vol. 151, pp. 682-683, 1966. [8] J. Pearson, “The orbital tower: a space craft launcher using the Earth’s rotational energy,” Acta Astronautica, vol. 2, pp.785-799, 1975. [9] Clarke. A. C., The Fountains of Paradise , Harcourt Brace Jovanovich, New York, 1979. [10] Edwards, B. C., “Design and deployment of a space elevator,” Acta Astronautica , vol. 47, no.10, pp.735-744, 2000. [11] Metal Working: Sheet Forming, ASM Handbook , volume 14B, 2006. [12] Benaroya, H., Mechanical Vibration: Analys is, Uncertainties and control. New York: Marcel Dekker Inc, 2004. [13] Yakobson, B. I. and Smalley R. E., “Fullerene nanotubes: C1,000,000 and Beyond,” American Scientist , vol.85, pp.324-337, 1997. [14] Perkins, N. C. and Mote, C. D. “Three-dimensional vibr ation of travelling elastic cables,” Journal of Sound and Vibration , vol. 114(2), pp. 325-340, 1987. [15] Han, S. M. and Benaroya, H. “Vib ration of a compliant tower in three- dimensions,” Journal of Sound and Vibration , vol. 250(4), pp. 675-709, 2002. Texas Tech University, Vijaya Kaithi, December 2008. 87 [16] Schutz, B., Gravity from the ground up , Cambridge University Press, 2003. [17] Schad, J., Physical Science: A Unified Approach, Brooks Cole, 1995. [18] MatLab – Signal Processing Toolbox, www.mathworks.com , 2006. [19] Kirkpatrick, L. D. and Wheeler, G. F. Physics: A World View , Harcourt Brace College Publishers, 3rd edition, 1998. Texas Tech University, Vijaya Kaithi, December 2008. 88 APPENDIX A MATLAB CODES Appendix A contains MATLAB codes used for obtaining different plots in previous Chapters A.1 taperedcrosssection.m clear; ro=6378e3; den=1300; g=9.81; rs=42164e3; strs=60e9:30e9:150e9; rf=152164e3; dr=(rf-ro)/100; for i=1:length(strs) r=[ro:dr:rf]’; h(i)=strs(i)/(den*g); tr=exp(ro/h(i)+ro^4/2/h(i)/rs^3-ro^2./r/h(i)-r.^2/2/h(i)/rs^3*ro^2) plot(r-ro,tr); hold on end A.2 mpovercablem.m clear; ro=6378e3; den=1300; omega=(2*pi)/(24*60*60); GM=398600e9; g=9.81; h=[6e6 13e6 25e6]'; rs=42164e3; dR=1000e3; dr=100e3; Rf=[42164e3:dR:152164e3]'; L=Rf-ro; for j=1:length(h) Texas Tech University, Vijaya Kaithi, December 2008. 89 strs(j)=h(j)*den*g; for i=1:length(Rf) rf=Rf(i); r=[6378e3:dr:rf]'; A = exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3)))+((ro/h(j))+((ro^4 )/(2*h(j)*rs^3)))); A1 = exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3 )))+((ro/h(j))+((ro^4)/( 2*h(j)*rs^3))))./(r.^2); A2= r.*exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3)))+((ro/h(j))+((ro^4 )/(2*h(j)*rs^3)))); cm(i)=den*trapz(A)*dr; Af1(i)=den*trapz(A1)*dr; Af2(i)=den*trapz(A2)*dr; mp(i)=(((omega^2*Af2(i))-(GM*Af1(i)))/((GM/rf^2)-(rf *(omega^2)))); mpoverm(i)=abs(mp(i))/cm(i); end figure(2) plot(L,mpoverm); hold on end A.3 natperiodtolength.m clear; counter=1; cn=1; ro=6378e3; rs=35786e3+ro; rho=1300; g=9.81; h=[2e6 8e6 20e6]'; GM=398600e9; E=1e12; omega=(2*pi)/(24*60*60); Rff=162164e3; dR=(Rff-rs)/100; Rf=[rs+dR:dR:Rff]'; L=Rf-ro; n2=length(Rf); n1=length(h); Texas Tech University, Vijaya Kaithi, December 2008. 90 for i=1:n1 sig(i)=h(i)*rho*g for j=1:n2 rf=Rf(j); dr=(rf-ro)/50; r=[ro+dr:dr:rf]'; tr(i)=exp(0.776*ro/h(i)); c=length(r); As=tr(i)*5e-6; A=As*exp(-ro^2/h(i)*(1./r+r.^2/2/rs^3))./exp(-ro^2/h(i )*(1./rs+rs.^2/2/rs^3)); e=ones(c,1); Afn=spdiags([A],0,c,c); Afn2=Afn; Aprime=(((ro^2)./(h(i)*r.^2))-((r.*ro^2)/(h(i)*rs^3))); Q=spdiags([e -2*e e],-1:1,c,c)./(dr^2); Q(c,c-1)=2/dr^2; k1=Afn*Q; Q2=spdiags([-e e],[-1,1],c,c)./(2*dr); Q2(c,c-1)=0; Aprime2=spdiags([Aprime],0,c,c); Aprimefn=Aprime2*Afn; k2=Aprimefn*Q2; r1=[ro:dr:rf]'; A1=As*exp(-ro^2/h(i)*(1./r1+r1.^2/2/rs^3 ))./exp(-ro^2/h(i)*(1./rs+rs.^2/2/rs^3)); f=(1./r1.^2-r1/rs^3); M(i)=rho*trapz(A1)*dr; mp(i)=rho*trapz(A1.*f)*dr*(rs ^3*rf^2)/(rf^3-rs^3); mratio=mp(i)/M(i); m=Afn*eye(c); m(c,c)=((Afn(c,c))+(2*mp(i)/(rho*dr))+(mp(i)*Aprime(c)/rho)); k=-[k1+k2]*eye(c)*E/rho; [ev,w2]=eig(k,m); temp=[(diag(w2)) ev']; sd=sortrows(temp); fr=sort(sqrt(diag(w2))); Texas Tech University, Vijaya Kaithi, December 2008. 91 tu1(j)=2*pi/(fr(1)*3600); tu2(j)=2*pi/(fr(2)*3600); tu3(j)=2*pi/(fr(3)*3600); beta1(j)=sqrt(fr(1)^2*rho/E); beta2(j)=sqrt(fr(2)^2*rho/E); beta3(j)=sqrt(fr(3)^2*rho/E); kV=-[k1+k2]*eye(c)*sig(i)/rho; [Vev,Vw2]=eig(kV,m); Vtemp=[(diag(Vw2)) Vev']; Vsd=sortrows(Vtemp); Vfr=sort(sqrt(diag(Vw2))); tV1(j)=2*pi/(Vfr(1)*3600); tV2(j)=2*pi/(Vfr(2)*3600); tV3(j)=2*pi/(Vfr(3)*3600); end figure(1) plot(L,beta1,L,beta2,L,beta3); hold on figure(2) plot(L,tu1,L,tu2,L,tu3); hold on figure(3) plot(L,tV1,L,tV2,L,tV3); hold on end A.4 crosssection.m clear; ro=6378e3; den=1300; g=9.81; rs=42164e3; E=1e12; strs=60e9; L=144000e3; Texas Tech University, Vijaya Kaithi, December 2008. 92 rf=L+ro; dr=(rf-ro)/100; for j=1:length(strs) h(j)=strs(j)/(den*g); B(j)=sqrt(1+2*strs(j)/E); r=[ro:dr:rf]'; el=B(j)*r; As=exp(ro^2/h(j)/B(j)^2*(1./(ro)+(ro).^2/2/r s^3))./exp(ro^2/h(j)/B(j)^2*(1./(ro+B(j)*(rs- ro))+(ro+B(j)*(rs-ro))^2/2/rs^3)); A = As*exp(ro^2/h(j)/B(j)^2*(1./(ro+B(j)*(rs-ro))+(ro+B(j)*(rs-ro))^2/2/rs^3))./exp(ro^2/h(j)/B(j)^2*(1./(ro+ B(j)*(r-ro))+(ro+B(j)*(r-ro)).^2/2/rs^3)); Aor=exp(ro/h(j)+ro^4/2/h(j)/rs^3-ro^2./r/h(j)-r.^2/2/h(j)/rs^3*ro^2); plot(el,A,el,Aor); hold on end A.5 pointmass.m clear; ro=6378e3; den=1300; E=1e12; g=9.81; GM=398600e9; strs=60e9; omega=(2*pi)/(24*60*60); rs=42164e3; dR=1000e3; dr=100e3; Rf=[42164e3:dR:152164e3]'; L=Rf-ro; for j=1:length(strs) h(j)=strs(j)/(den*g); B(j)=sqrt(1+2*strs(j)/E); l=B(j)*L; for i=1:length(Rf) Texas Tech University, Vijaya Kaithi, December 2008. 93 rf=Rf(i); r=[6378e3:dr:rf]'; As=exp(ro^2/h(j)/B(j)^2*(1./(ro)+(ro).^2/2/r s^3))./exp(ro^2/h(j)/B(j)^2*(1./(ro+B(j)*(rs- ro))+(ro+B(j)*(rs-ro))^2/2/rs^3)); A = As*exp(ro^2/h(j)/B(j)^2*(1./(ro+B(j)*(rs-ro))+(ro+B(j)*(rs-ro))^2/2/rs^3))./exp(ro^2/h(j)/B(j)^2*(1./(ro+ B(j)*(r-ro))+(ro+B(j)*(r-ro)).^2/2/rs^3)); cm(i)=den*trapz(A)*dr; Mp(i)=B(j)*strs(j)/omega^2/((ro+B(j)*L(i))-rs^3/(ro+B (j)*L(i))^2)*... exp(ro^2/h(j)/B(j)^2*(1./(ro+B(j)* (rs-ro))+(ro+B(j)*(rs-ro))^2/2/rs^3))*... (exp(-ro^2/h(j)/B(j)^ 2*(1./(ro+B(j)*L(i))+(ro+B(j)*L(i))^2/2/rs^3))-exp(- ro^2/h(j)/B(j)^2*(1/ro+ro^2/2/rs^3))); Mpoverm(i)=abs(Mp(i))/cm(i); Aor = exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3)))+((ro/h(j))+((ro^4 )/(2*h(j)*rs^3)))); A1 = exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3 )))+((ro/h(j))+((ro^4)/( 2*h(j)*rs^3))))./(r.^2); A2= r.*exp((((-ro^2)./(r*h(j)))-((ro^2*r.^2)/(2*h(j)*rs^3)))+((ro/h(j))+((ro^4 )/(2*h(j)*rs^3)))); cm(i)=den*trapz(Aor)*dr; Af1(i)=den*trapz(A1)*dr; Af2(i)=den*trapz(A2)*dr; mp(i)=(((omega^2*Af2(i))-(GM*Af1(i)))/((GM/rf^2)-(rf *(omega^2)))); mpoverm(i)=abs(mp(i))/cm(i); end figure(1) plot(l,Mpoverm,l,mpoverm) hold on end A.6 accelpromoon.m clear; ro=6378e3; rs=35786e3+ro; G=6.67e-11; Mm=0.07349e24; Me=5.9742e24; Dem=3.84467e8; Sem=(Mm*Dem)/(Me+Mm); Texas Tech University, Vijaya Kaithi, December 2008. 94 omega2=sqrt(G*(Mm+Me)/Dem^3); omega=(2*pi)/(24*60*60); double(omega); double(omega2); t=0:1000:36e4; r1=ro; r2=rs; r3=100e6; a1=(Dem*((cos(28.65*pi/180)*cos(t.*omega).*c os(t.*omega2))+(sin(t.*omega2).*sin(t.* omega)))-r1).^2; b1=Dem^2*(cos(t.*omega2).*sin(28.65*pi/180)).^2; c1=Dem^2*((cos(28.65*pi/180)*cos(t.*omega2).*sin(t.*omega))-(sin(t.*omega2).*cos(t.*omega))).^2; Rem1=sqrt((a1+b1+c1)); pu11=((cos(28.65*pi/180)*cos(t.*omega).*cos(t.*omega2))+(sin(t.*omega2).*sin(t.*omega))).*((G*Mm*Dem./Rem1.^3)-(Sem*omega2^2)); pu12=(G*Mm*r1./Rem1.^3); pu1=pu11-pu12; a2=(Dem*((cos(28.65*pi/180)*cos(t.*omega).*c os(t.*omega2))+(sin(t.*omega2).*sin(t.* omega)))-r2).^2; b2=Dem^2*(cos(t.*omega2).*sin(28.65*pi/180)).^2; c2=Dem^2*((cos(28.65*pi/180)*cos(t.*omega2).*sin(t.*omega))-(sin(t.*omega2).*cos(t.*omega))).^2; Rem2=sqrt((a2+b2+c2)); pu21=((cos(28.65*pi/180)*cos(t.*omega).*cos(t.*omega2))+(sin(t.*omega2).*sin(t.*ome ga))).*((G*Mm*Dem./Rem2.^3)-(Sem*omega2^2)); pu22=(G*Mm*r2./Rem2.^3); pu2=pu21-pu22; a3=(Dem*((cos(28.65*pi/180)*cos(t.*omega).*c os(t.*omega2))+(sin(t.*omega2).*sin(t.* omega)))-r3).^2; b3=Dem^2*(cos(t.*omega2).*sin(28.65*pi/180)).^2; c3=Dem^2*((cos(28.65*pi/180)*cos(t.*omega2).*sin(t.*omega))-(sin(t.*omega2).*cos(t.*omega))).^2; Rem3=sqrt((a3+b3+c3)); pu31=((cos(28.65*pi/180)*cos(t.*omega).*cos(t.*omega2))+(sin(t.*omega2).*sin(t.*omega))).*((G*Mm*Dem./Rem3.^3)-(Sem*omega2^2)); pu32=(G*Mm*r3./Rem3.^3); pu3=pu31-pu32; pv1=(sin(28.65*pi/180)*cos(t.*omega2)).*((G*Mm*Dem./Rem1.^3)-(Sem*omega2^2)); pv2=(sin(28.65*pi/180)*cos(t.*omega2)).*((G*Mm*Dem./Rem2.^3)-(Sem*omega2^2)); pv3=(sin(28.65*pi/180)*cos(t.*omega2)).*((G*Mm*Dem./Rem3.^3)-(Sem*omega2^2)); Texas Tech University, Vijaya Kaithi, December 2008. 95 pw1=((cos(28.65*pi/180)*sin(t.*omega).*cos(t.*omega2))- (sin(t.*omega2).*cos(t.*omega))).*((G *Mm*Dem./Rem1.^3)-(Sem*omega2^2)); pw2=((cos(28.65*pi/180)*sin(t.*omega).*cos(t.*omega2))-(sin(t.*omega2).*cos(t.*omega))).*((G *Mm*Dem./Rem2.^3)-(Sem*omega2^2)); pw3=((cos(28.65*pi/180)*sin(t.*omega).*cos(t.*omega2))-(sin(t.*omega2).*cos(t.*omega))).*((G *Mm*Dem./Rem3.^3)-(Sem*omega2^2)); subplot(3,1,1);plot(t/ 3600,pu1,t/3600,pu2,t/3600,pu3); subplot(3,1,2);plot(t/ 3600,pv1,t/3600,pv2,t/3600,pv3); subplot(3,1,3);plot(t/ 3600,pw1,t/3600,pw2,t/3600,pw3); A.7 sunpsd.m clear; t=0:10000:3e7; r=100e6; %properties of the earth ro=6378e3; rs=35786e3+ro; G=6.67e-11;%universal gr avitational costant Me=5.9742e24;%mass of earth Ms=1.989e30;%mass of sun Des=1.496e11;%distance from moon to earth centre omega=(2*pi)/(24*60*60);%rota tional rate of earth thetas=23.5*pi/180;%the angle between plane of earths equator and earth sun orbital plane omegaes=sqrt(G*(Ms+Me)/Des^3);%angular velocity of earth sun system Ses=(Ms*Des)/(Me+Ms);%distance from earth to centre of mass d1=(cos(thetas).*cos(omega*t).*cos(omegaes *t))+(sin(omegaes*t).*sin(omega*t));%ith component of vector drawn from earh to sun centre d2=(cos(omegaes*t).*sin(thetas));%jth com ponent of vector drawn from earh to sun centre d3=((cos(thetas)*cos(omegaes*t).*sin(omega*t))-(sin(omegaes*t).*cos(omega*t)));%kth component of vector drawn from earh to sun centre Res=sqrt((Des*d1-(r)).^2+Des^2*d2.^2+Des^2*d3.^2);%distance from elevator to sun centre Texas Tech University, Vijaya Kaithi, December 2008. 96 pus=((G*Ms*Des./Res.^3)-(Ses*omegaes^2)-(G*Ms*(r)./Res.^3)).*d1;%differential tidal accelerations due to sun in radial direction pvs=((G*Ms*Des./Res.^3)-(Ses*omegaes^2)).*d2;%differential tidal accelerations due to sun in meridional direction pws=((G*Ms*Des./Res.^3)-(Ses*omegaes^2)).*d3; %differential tidal accelerations due to sun in equitorial direction N=2^10 subplot(3,1,1); psd(pus,N,1/(t(2)-t(1))); subplot(3,1,2); psd(pvs,N,1/(t(2)-t(1))); subplot(3,1,3); psd(pws,N,1/(t(2)-t(1))); A.8 newresponse3.m clear; global counter cn counter=1; cn=1; ro=6378e3; rs=35786e3+ro; rho=1300; g=9.81; sig=60e9; GM=398600e9; E=1e12; omega=(2*pi)/(24*60*60); Rff=11.6378e7; Ri=56.378e6; dR=(Rff-Ri)/6; Rf=[Ri+dR:dR:Rff]'; n2=length(Rf); n1=length(sig); Texas Tech University, Vijaya Kaithi, December 2008. 97 for i=1 h(i)=sig(i)/(rho*g); for j=1:n2 rf=Rf(j); dr=(rf-ro)/20; r=[ro+dr:dr:rf]'; tr(i)=exp(0.776*ro/h(i)); c=length(r); As=tr(i)*5e-8; A=As*exp(-ro^2/h(i)*( 1./r+r.^2/2/rs^3))./exp(-ro^2/h(i)*(1./rs+rs.^2/2/rs^3));%As=1 e=ones(c,1); Afn=spdiags([A],0,c,c); Afn2=Afn; Aprime=(((ro^2)./(h(i)*r.^2))-((r.*ro^2)/(h(i)*rs^3))); Q=spdiags([e -2*e e],-1:1,c,c)./(dr^2); Q(c,c-1)=2/dr^2; k1=Afn*Q; Q2=spdiags([-e e],[-1,1],c,c)./(2*dr); Q2(c,c-1)=0; Aprime2=spdiags([Aprime],0,c,c); Aprimefn=Aprime2*Afn; k2=Aprimefn*Q2; r1=[ro:dr:rf]'; A1=As*exp(-ro^2/h(i)*(1./r1+r1.^2/2/rs^3 ))./exp(-ro^2/h(i)*(1./rs+rs.^2/2/rs^3)); f=(1./r1.^2-r1/rs^3); M(i)=rho*trapz(A1)*dr; mp(i)=rho*trapz(A1.*f)*dr*(rs ^3*rf^2)/(rf^3-rs^3); mratio=mp(i)/M(i); m=Afn*eye(c); m(c,c)=((Afn(c,c))+(2*mp(i)/(rho*dr))+(mp(i)*Aprime(c)/rho)); k=-[k1+k2]*eye(c)*E/rho; [ev,w2]=eig(k,m); temp=[(diag(w2)) ev']; sd=sortrows(temp); fr=sort(sqrt(diag(w2))); tu1(i,j)=2*pi/(fr(1) *3600); %Longitudinal Texas Tech University, Vijaya Kaithi, December 2008. 98 tu2(i,j)=2*pi/ (fr(2)*3600); tu3(i,j)=2*pi/ (fr(3)*3600); D=[zeros(c,c) eye(c);inv(-m)*k zeros(c,c)]; t=[0:0.1:200]*3600; Yo=zeros(2*c,1); option=odeset('RelTol', 1e-8, 'AbsTOl', 1e-8); [t,U]=ode45(@taperufun4,t,Y o,[],D,m,c,r,rho,A); utip=U(:,c); figure(3) subplot(3,2,j); plot(t/3600,U(:,end)); figure(1) subplot(3,2,j); plot(t/3600,utip); kV=-[k1+k2]*eye(c)*sig(i)/rho; ApAseon=diag(Aprime); Kseon=sig(i)/rho*(ApAseon*Q2+Q); Gseon=[zeros(c,c) eye(c);Kseon zeros(c,c)]; [Vev,Vw2]=eig(kV,m); Vtemp=[(diag(Vw2)) Vev']; Vsd=sortrows(Vtemp); Vfr=sort(sqrt(diag(Vw2))); tV1(i,j)=2*pi/(Vfr(1)*3600); %Transverse tV2(i,j)=2*pi/(Vfr(2)*3600); tV3(i,j)=2*pi/(Vfr(3)*3600); G=[zeros(c,c) eye(c);inv(-m)*kV zeros(c,c)]; t=[0:0.1:200]*3600; Yo=zeros(2*c,1); [t,V]=ode45(@tapervfun2,t,Y o,[],G,m,c,r,rho,A); Vtip=V(:,c); %eval(['save data',num2str(i),num2str(j),' t r V Vtip;']) figure(4) subplot(3,2,j); plot(t/3600,V(:,end)); figure(2) subplot(3,2,j); plot(t/3600,Vtip); end Texas Tech University, Vijaya Kaithi, December 2008. 99 end A.9 taperufun4.m function dot=taperufun4(t,Y,D,m,c,r,rho,A) global counter cn [pu pv pw]=tidalf(r,t); dot=(D*Y)+[zeros(c,1);inv(m)*pu.*A]; counter=counter+1; if counter==1000*cn; t/3600 cn=cn+1; end A.10 tapervfun2.m function dot=tapervfun2(t,Y,G,m,c,r,rho,A) global counter cn [pu pv pw]=tidalf(r,t); dot=(G*Y)+[zeros(c,1);inv(m)*pv.*A]; counter=counter+1; if counter==1000*cn; t/3600 cn=cn+1; end A.11 threedresponse.m clear global counter cn N h Q1 Q2 global go ro rs X L omega m Mp sigma E r ho hh Ap Al Mo Mm Ms G rmoon rsun force global wfu wfv wfw N=20; %number of nodes %properties of the Earth omega=7.272205216643040e-005; facs=1; G=6.672000000000000e-011; Texas Tech University, Vijaya Kaithi, December 2008. 100 Mo=5.973600000000001e+024; Mm=0.07349e24; Ms=1.989e30; ro=6378000; go=G*Mo/ro^2; rs=(G*Mo/omega^2)^(1/3); rmoon=3.84467e8; rsun=1.496e11; %properties of the cable E=1e12; rho=1300; sigma=60e9/facs; L=8.2e7; %spatial dimension h=L/N; X=[h:h:L]'; rs=(G*Mo/omega^2)^(1/3); rl=ro+L; hh=sigma/rho/go; B=sqrt(1+2*sigma/E); As=0.115*1e-6; Mp=As*B*sigma/omega^2/( (ro+B*L)-rs^3/(ro+B*L)^2)*... exp(ro^2/hh/B^2*(1./(ro+B*(rs-r o))+(ro+B*(rs-ro)) ^2/2/rs^3))*... (exp(-ro^2/hh/B^2*(1./(ro+B *L)+(ro+B*L)^2/2/rs^3))-exp(- ro^2/hh/B^2*(1/ro+ro^2/2/rs^3))); A=As*exp(-ro^2/hh/B^2*(1./(ro+B*X)+(ro+B*X).^2/2/rs^3))... /exp(-ro^2/hh/B^2*(1./(ro+B*(rs-r o))+(ro+B*(rs-ro))^2/2/rs^3)); Ao=As*exp(-ro^2/hh/B^2*(1/ro+ro^2/2/rs^3))... /exp(-ro^2/hh/B^2*(1./(ro+B*(rs-r o))+(ro+B*(rs-ro))^2/2/rs^3)); %cross-sectional area at the surface taper=max(A)/Ao; Al=A(end); %area at th e end of the cable Ap=ApA(X); % A'/A along the cable %necessary matrices e=ones(N,1); Q1=spdiags([-e e],[-1 1], N,N)./(2*h); Q2=spdiags([e -2*e e],-1:1,N,N)./(h^2); Texas Tech University, Vijaya Kaithi, December 2008. 101 force=1; %time integration tf=3600*200; %final time tint=3600*0.1;% time at which the values ar e recorded qn=1; %number of times that the data is saved to disk %initial diplacement vc=0; wc=0; % Vo=vc*X; Wo=wc*X; % Vop=vc; Wop=wc; % Vopp=zeros(size(X)); Wopp=zeros(size(X)); % %STATIC %static configuration Qu1=Q1; Qu1(N,N-1)=0; Qu2=Q2; Qu2(N,N-1)=2/h^2; Qu2(N,N)=-2/h^2; uepl=Mp/E/Al*((ro+L)*omega^2-go*ro^2/(ro+L)^2); r=ro+X; ue=inv(Qu2+diag(ApA(X))*Qu1)*(rho/ E*(go*ro^2./r.^2-r.*omega^2)... -[zeros(N-1,1);2/h*uepl] -[zeros(N-1,1);ApA(L)*uepl]); for guess=1:30 tmp=Mp/E/Al*((ro+L+ue(end))*om ega^2-go*ro^2/(ro+L+ue(end))^2); r=ro+X+ue; Root=roots([0.5 1.5 1 -tmp]); I=find(Root>0); uepl=Root(I); uep=Q1*ue; uep(end)=uepl; ue=inv(Qu2)*((rho/E* go*ro^2*(1./r.^2-r/rs^3)... -ApA(X).*uep.*(1+1.5*uep+ 0.5*uep.^2))./(1+3*uep+1.5*uep.^2)... -[zeros(N-1,1);2/h*uepl]); end IC(:,1)=[ue;zeros(N,1);V o;zeros(N,1);Wo;Vo]; Yo=IC(:,1); %DYNAMIC Ys=Yo'; Texas Tech University, Vijaya Kaithi, December 2008. 102 ts=0; save prac ts Yo Ys options=odeset('RelTol',1e-10,'AbsTol', 1e-10); int=tf/qn; PLOT=1; counter=0; cn=1; for q=1:qn; tspan=[int*(q-1):tint:int*q]'; load prac [t,Y]=ode45('mainfuns',tsp an,Yo); % dynamic displacements ts=[ts;t(2:size(t,1))]; NN=size(t,1); tmpY=Y(2:size(Y,1),:); Yo=Y(NN,:); Ys=[Ys;tmpY]; save prac ts Yo Ys if PLOT==1; figure(1) subplot(3,1,1) plot(ts/3600,Ys(:,N)-ue(N),'b') grid subplot(3,1,2) plot(ts/3600,Ys(:,3*N),'b') grid subplot(3,1,3) plot(ts/3600,Ys(:,5*N),'b') grid figure(2) subplot(3,1,1) plot(ts/3600,Ys(:,2*N)) grid subplot(3,1,2) plot(ts/3600,Ys(:,4*N)) grid subplot(3,1,3) plot(ts/3600,Ys(:,6*N)) grid pause(1) end end Texas Tech University, Vijaya Kaithi, December 2008. 103 save 3Dcase2 ts Yo Ys A.12 mainfuns.m function dot=mainfuns(t,Y) global counter cn N h Q1 Q2 go L ro E X omega rho Mp Al Ap G Mm Ms Mo rmoon rsun force %force due to sun and moon pu=zeros(N,1); pv=zeros(N,1); pw=zeros(N,1); if force==1 w=2*pi/24/3600; omm=sqrt(G*(Mm+Mo)/rmoon.^3); thetam=28.65*pi/180; sm=Mm*rmoon/(Mo+Mm); c1=cos(thetam).*cos(omm*t).*cos(w*t)+sin(omm*t).*sin(w*t); c2=-sin(thetam).*cos(omm*t); c3=cos(thetam).*cos(omm*t).*sin (w*t)-sin(omm*t).*cos(w*t); Rm=sqrt((rmoon*c1-(ro+X)).^2+r moon^2*c2.^2+rmoon^2*c3.^2); % pum=-G*Mm./Rm.^3.*(ro+X)+(G*M m./Rm.^3*rmoon-omm^2*sm).*c1; pvm=(G*Mm./Rm.^3*rmoon-omm^2*sm).*c2; pwm=(G*Mm./Rm.^3*rmoon-omm^2*sm).*c3; thetas=23.5*pi/180; oms=sqrt(G*(Ms+Mo)/rsun.^3); ss=Ms*rsun/(Mo+Ms); d1=cos(thetas).*cos(oms*t).*cos(w*t)+sin(oms*t).*sin(w*t); d2=-sin(thetas).*cos(oms*t); d3=cos(thetas).*cos(oms*t).*sin (w*t)-sin(oms*t).*cos(w*t); Rs=sqrt((rsun*d1-(ro+X)).^2+rsun^2*d2.^2+rsun^2*d3.^2); pus=-G*Ms./Rs.^3.*(ro+X)+(G*Ms./Rs.^3*rsun-oms^2*ss).*d1; pvs=(G*Ms./Rs.^3*rsun-oms^2*ss).*d2; pws=(G*Ms./Rs.^3*rsun-oms^2*ss).*d3; pv=pvm+pvs; pu=pum+pus; pw=pwm+pws end u=Y(1:N); v=Y(2*N+1:3*N); w=Y(4*N+1:5*N); ud=Y(N+1:2*N); vd=Y(3*N+1:4*N); wd=Y(5*N+1:6*N); up=Q1*u; upp=Q2*u; vp=Q1*v; vpp=Q2*v; wp=Q1*w; wpp=Q2*w; r=sqrt((ro+X+u).^2+v.^2+w.^2);% Texas Tech University, Vijaya Kaithi, December 2008. 104 ru=ro+X+u; % Apl=ApA(L); strain=up+0.5*(up.^2+vp.^2+wp.^2); % strainp=upp+up.*upp+vp.*vpp+wp.*wpp; % udd=-2*omega*wd+ru*omega^2-go*ro^2*ru./r.^3.+E/rho*(strainp.*(1+ up)+strain.*(upp+Ap.*(1+up)))+pu;% vdd=-go*ro^2.*v./r.^3+E/rho*(stra inp.*vp+strain.*(vpp+Ap.*vp))+pv; wdd=2*omega*ud+w*omega^2-go*ro^2.*w./r.^3+E/rho.*(strainp.*w p+strain.*(wpp+Ap.*wp))+pw; ul=u(end); vl=v(end); wl=w(end); udl=ud(end); vdl=vd(end); wdl=wd(end); ulm=u(end-1); vlm=v(end-1); wlm=w(end-1); rl=r(end); % rul=ro+L+u(end); % last=[udd(end-1); vdd(end -1); wdd(end-1)]; F=1; index=1; for JJJ=1:5 uddl=last(1); vddl=l ast(2); wddl=last(3); TMP1=-Mp/E/Al*(uddl+2*wdl*omega-rul*omega^2+go*ro^2*rul/rl^3); TMP2=-Mp/E/Al*(vddl+go*ro^2*vl/rl^3); TMP3=-Mp/E/Al*(wddl-2*udl*ome ga-wl*omega^2+go*ro^2*wl/rl^3); Root=roots([2 1 0 -(TMP1^2+TMP2^2+TMP3^2)]); I=find(Root>0); epsilon=Root(I); upl=TMP1/epsilon-1; uppl=2*(-ul+ulm)/h^2+2/h*upl; vpl=TMP2/epsilon; vppl=2*(-vl+vlm)/h^2+2/h*vpl; wpl=TMP3/epsilon; wppl=2*(-wl+wlm)/h^2+2/h*wpl; epsilonp=uppl+upl *uppl+vpl*vppl+wpl*wppl; F=[uddl+2*wdl*omega-rul*omega^2-E/rho*(epsilonp*(1+upl)+epsilon*(uppl+Apl *(1+upl)))+go*ro^2*r ul/rl^3-pu(end); vddl-E/rho*(epsilonp*vpl+epsil on*(vppl+Apl*vpl))+go*ro^2*vl/rl^3-pv(end); wddl-2*udl*omega-wl*omega^2-E/rho*(epsilonp*wpl+epsilon*(wppl+Apl *wpl))+go*ro^2*wl/rl^3-pw(end)]; dEdu=-2*Mp/E/Al*TMP 1/(6*epsilon^2+2*epsilon); dEdv=-2*Mp/E/Al*TMP 2/(6*epsilon^2+2*epsilon); Texas Tech University, Vijaya Kaithi, December 2008. 105 dEdw=-2*Mp/E/Al*TMP 3/(6*epsilon^2+2*epsilon); tmp=(-Mp/E/Al/epsilon); dupdu=tmp-TMP1/epsilon^2*dEdu; duppdu=2/h*dupdu; dvpdu=-TMP2/epsilon^2*dEdu; dvppdu=2/h*dvpdu; dwpdu=-TMP3/epsilon^2*dEdu; dwppdu=2/h*dwpdu; dupdv=-TMP1/epsilon^2*dEdv; duppdv=2/h*dupdv; dvpdv=tmp-TMP2/epsilon^2*dEdv; dvppdv=2/h*dvpdv; dwpdv=-TMP3/epsilon^2*dEdv; dwppdv=2/h*dwpdv; dupdw=-TMP1/epsilon^2*dEdw; duppdw=2/h*dupdw; dvpdw=-TMP2/epsilon^2*dEdw; dvppdw=2/h*dvpdw; dwpdw=tmp-TMP3/epsilon^2*dEdw; dwppdw=2/h*dwpdw; dEpdu=duppdu+dupdu*uppl+duppdu*upl+dvpdu*vppl+dvppdu*vpl+dwpdu*wppl+dwppdu*wpl; dEpdv=duppdv+dupdv*uppl+duppdv*upl+dvpdv*vppl+dvppdv*vpl+dwpdv*wppl+dwppdv*wpl; dEpdw=duppdw+dupdw*uppl+duppdw*upl+ dvpdw*vppl+dvppdw*vpl+dwpdw*wppl+d wppdw*wpl; J=[1-E/rho*(dEpdu*(1+upl)+epsilonp*dupdu+dE du*(uppl+Apl*(1+upl))+epsilon*(duppdu+A pl*dupdu))... -E/rho*(dEpdv*(1+upl)+epsilonp*dupdv+dE dv*(uppl+Apl*(1+upl))+epsilon*(duppdv+A pl*dupdv))... -E/rho*(dEpdw*(1+upl)+epsilonp*dupdw+d Edw*(uppl+Apl*(1+upl))+epsilon*(duppdw +Apl*dupdw)); -E/rho*(dEpdu*vpl+epsilonp*dvpdu+dEdu*(vppl+A pl*vpl)+epsilon*(dvppdu+Apl*dvpd u)) ... 1-E/rho*(dEpdv*vpl+epsilonp*dvpdv+dEdv*(vppl+A pl*vpl)+epsilon*(dvppdv+Apl*dvpd v)) ... Texas Tech University, Vijaya Kaithi, December 2008. 106 - E/rho*(dEpdw*vpl+epsilonp*dvpdw+dEdw*( vppl+Apl*vpl)+eps ilon*(dvppdw+Apl*dv pdw)); -E/rho*(dEpdu*wpl+epsilonp*dwpdu+dEdu*(wppl+Apl*wpl)+epsilon*(dwppdu+Apl*dwpdu)) ... -E/rho*(dEpdv*wpl+epsilonp*dwpdv+dEdv*(wppl+Apl*wpl)+epsilon*(dwppdv+Apl*dwpdv)) ... 1-E/rho*(dEpdw*wpl+epsilonp*dwpdw+dEdw*( wppl+Apl*wpl)+ep silon*(dwppdw+Apl* dwpdw))]; last=last-inv(J)*F; index=index+1; end udd(end)=last(1); vdd(end)=last(2); wdd(end)=last(3); dot=[ud; udd; vd; vdd;wd;wdd]; counter=counter+1; if counter==1000*cn; t/3600 cn=cn+1; end A.13 APA.m function out=ApA(X) global ro rs hh global sigma E B=sqrt(1+2*sigma/E); out=ro^2/hh/B*(1./(ro+B*X).^2-(ro+B*X)/rs^3); A.14 modalboth.m clear; tic r0=6378e3; rs=35786e3+r0; rho=1300; g=9.81; sig=60*1e9; Texas Tech University, Vijaya Kaithi, December 2008. 107 h=sig/rho/g; GM=398600e9; E=1e12; omega=(2*pi)/(24*60*60); Ri=5e7+r0; Rff=10e7+r0; dR=(Rff-Ri)/5000; Rf=[[Ri:20*dR:7e7+r0] [7.4e7+r0+dR:dR:7.9e7+r0] [7.9e7+r0+dR:20*dR: Rff]]'; length(Rf) n2=length(Rf); n3=8; %number of modes nq=10; %number of fourier expansion for Q for j=1:n2 %j=1; rf=Rf(j); L(j)=rf-r0; c=20; dr=(rf-r0)/c; r=[r0+dr:dr:rf]'; tr=exp(0.776*r0/h); A=exp((((-r0^2)./(r*h))-((r0^2*r.^2)/(2*h*rs^3)))+((r0/h)+((r0^4)/(2*h*rs^3)))); A=A/max(A); e=ones(c,1); Afn=spdiags([A],0,c,c); Aprime=(((r0^2)./(h*r.^2))-((r.*r0^2)/(h*rs^3))); %A'/A Q=spdiags([e -2*e e],-1:1,c,c)./(dr^2); Q(c,c-1)=2/dr^2; k1=Afn*Q; Q2=spdiags([-e e],[-1,1],c,c)./(2*dr); Q2(c,c-1)=0; Aprime2=spdiags([Aprime],0,c,c); Aprimefn=Aprime2*Afn; k2=Aprimefn*Q2; As=1; r1=[r0:dr:rf]'; A1=As*exp(-r0^2/h*(1./r1+r1.^2/2/rs^3 ))./exp(-r0^2/h*(1./rs+rs.^2/2/rs^3)); Texas Tech University, Vijaya Kaithi, December 2008. 108 f=(1./r1.^2-r1/rs^3); M=rho*trapz(A1)*dr; mp=rho*trapz(A1.*f)*dr*(rs^3*r f^2)/(rf^3-rs^3); % mratio=mp/M; Aratio=trapz(A1)/A1(end); m=Afn*eye(c); m(c,c)=((Afn(c,c))+(2*mp/(r ho*dr))+(mp*Aprime(c)/rho)); k=-[k1+k2]*eye(c)*E/rho; KU=k; mU=m; for p=1:c-1 fac=KU(p,p+1)/KU(p+1, p); mU(p+1, :)=mU(p+1,:)*fac; KU(p+1, :)=KU(p+1,:)*fac; end KU=round(1e12*KU)/1e12; [V,D]=eig(KU,mU); temp=[(diag(D)) V']; sd=sortrows(temp); fr=sort(sqrt(diag(D))); periodu(j,1:n3)=2*pi./(fr(1:n3)'*3600); wn(1:n3)=fr(1:n3); U=sd(1:n3,2:end)'; kV=-[k1+k2]*eye(c)*sig/rho; ApAseon=diag(Aprime); KV=kV; mV=m; for p=1:c-1 fac=KV(p,p+1)/KV(p+1, p); mV(p+1, :)=mV(p+1,:)*fac; KV(p+1, :)=KV(p+1,:)*fac; end KV=round(1e12*KV)/1e12; [Vev,Vw2]=eig(KV,mV); Vtemp=[(diag(Vw2)) Vev']; Vsd=sortrows(Vtemp); Vfr=sort(sqrt(diag(Vw2))); periodv(j,1:n3)=2*pi./(Vfr(1 :n3)'*3600); %Transverse Texas Tech University, Vijaya Kaithi, December 2008. 109 wVn(1:n3)=Vfr(1:n3); Y(:,1:n3)=Vsd(1:n3,2:end)'; %these are same as U1, U2 and U3 tf=25; dt=0.1; x=r-r0; t=[0:dt:tf]'*3600; [pum pvm pwm pus pvs pws]=tidalfnew(x,t); dx=x(2)-x(1); %CAlculating maximum displacement AFN=repmat(diag(Afn)', size(pum,1),1); Qum=(pum.*AFN)*Y; Qus=(pus.*AFN)*Y; Qvm=(pvm.*AFN)*Y; Qvs=(pvs.*AFN)*Y; wm=2*pi/(25*3600); ws=2*pi/(24*3600); fourierQum=zeros(n3,size(t,1)); fourierQus=zeros(n3,size(t,1)); fourierQvm=zeros(n3,size(t,1)); fourierQvs=zeros(n3,size(t,1)); Amu=zeros(n3, nq); Asu=zeros(n3, nq); for k=1:nq %ith mode, k number of fourier expansions in forcing terms Amu(:,k)=(trapz(Qum.*repmat(cos((k -1)*wm*t),1,n3)))'/trapz(cos((k-1)*wm*t).^2); Asu(:,k)=(trapz(Qus.*repmat(cos((k-1)*ws*t),1,n3)))'/trapz(cos((k-1)*ws*t).^2); Amv(:,k)=(trapz(Qvm.*repmat(cos((k -1)*wm*t),1,n3)))'/trapz(cos((k-1)*wm*t).^2); Asv(:,k)=(trapz(Qvs.*repmat(cos((k-1)*ws*t),1,n3)))'/trapz(cos((k-1)*ws*t).^2); fourierQum=fourierQum+Amu (:,k)*cos((k-1)*wm*t'); fourierQus=fourierQus+Asu(:,k)*cos((k-1)*ws*t'); fourierQvm=fourierQvm+Amv (:,k)*cos((k-1)*wm*t'); fourierQvs=fourierQvs+Asv(:,k)*cos((k-1)*ws*t'); end wmk=[0:nq-1]*wm; wsk=[0:nq-1]*ws; [WMK WN]=meshgrid(wmk, wn); [WSK WN]=meshgrid(wsk, wn); Texas Tech University, Vijaya Kaithi, December 2008. 110 %[WMK WVN]=meshgrid(wmk, wVn); [WSK WVN]=meshgrid(wsk, wVn); betau=Amu./(WN.^2-WMK.^2); alphau=Asu./(WN.^2-WSK.^2); betav=Amv./(WVN.^2-WMK.^2); alphav=Asv./(WVN.^2-WSK.^2); ampetau=(sum(abs(betau'))+sum(abs(alphau')))'; ampetav=(sum(abs(betav'))+sum(abs(alphav')))'; maxuL(j)=abs(Y(end,:))*ampetau; maxvL(j)=abs(Y(end,:))*ampetav; if rem(j,100)==1 j length(Rf) end end figure(1) subplot(2,1,1) plot(L,log10(maxuL)); subplot(2,1,2) plot(L,log10(maxvL)); toc Texas Tech University, Vijaya Kaithi, December 2008. 111APPENDIX B FINITE DIFFERENCE METHOD Let us find the discrete form of the continuous model for the elevator equation given in Eqn 2.18. Let’s assume that the elevator is divided into n number of elements as shown in figure B.1. The equation of motion of space elevator at ith node is given by, ሻ ሺB. 1ሻ ܣߩ ௜ݕሷ௜ൌ൫ ܣ ܪ ሺݔሻݕ௜′൯′൅݂ ௜ሺݐሻ ሺ݅ ൌ 1 ݊ ݋ݐ with boundary conditions ݑ ଵሺݐሻൌ0 , ܯ ௣ݕሷ௡൅ܣ ܪ ௡ݕ௡′ൌ0 , ݂௜ሺݐሻ indicates the tidal forces acti ng per unit length of the elevator. Eqn. B.1 can be written as, ݕሷ ௜ൌܪ ߩܣ௜ᇱ ܣ௜ݕ௜ᇱ൅ܪ ߩݕ௜ᇱᇱ൅݌ ௜ሺݐሻ, ሺB. 2ሻ where ݌௜ሺݐሻ is the tidal force per unit mass. For longitudinal motion, ݌௜ሺݐሻ will be replaced by the tidal acceleration in th e radial direction of the elevator ݌௨௜ሺݐሻ and for transverse motion ݌௜ሺݐሻ will be replaced by the tidal acceleration in the equatorial direction of the elevator ݌௨௜ሺݐሻ. We can write ݕ௜ᇱ and ݕ௜ᇱᇱ in terms of ݕ௜, ݕ௜ାଵ, ݕ௜ିଵ by using the finite difference formulae as, ݕ௜ᇱ ൌݕ௜ାଵെݕ ௜ିଵ 2∆ , ሺB. 3ሻ ݕᇱᇱൌݕ௜ାଵെ2 ݕ ௜൅ݕ ௜ିଵ ௜∆ଶ , where ∆ .݁.݅ ݐ݈݊݁݉݁݁ ݂݋ ݄ݐ݈݃݊݁ ݄݁ݐ ݏ݅ ሺ ݎ௜ାଵെݎ ௜ሻ. Substituting Eqn B.3 in boundary condit n at tip of the elevator in Eqn B.1 gives, io the ܯ ௣ݕሷ௡൅ܣܪ ௡ቀݕ௡ାଵ െݕ ௡ିଵ 2∆ቁ ൌ 0, ሺB. 4ሻ Texas Tech University, Vijaya Kaithi, December 2008. 112 From Eqn 2.23, ݕ௡ାଵ can be written as, ݕ ௡ାଵ ൌݕ ௡ିଵ െ2 ܯ ௣ݕሷ௡∆ ܣ ܪ ௡ ሺB. 5ሻ The last equation from the set of equa tions in B.2, i.e. the EOM at the nth node, can be written as ݕሷ ௡ൌܪ ߩܣ௡ᇱ ܣ௡ݕ௡ᇱ൅ܪ ߩݕ௡ᇱᇱ൅݌ ௡ሺݐሻ ሺB. 6ሻ Substituting Eqn. B.3 in Eqn. B.6 gives, ݕ ሷ௡ൌܪ ߩܣ௡ᇱ ܣ௡ቀݕ௡ାଵ െݕ ௡ିଵ 2∆ቁ൅ܪ ߩ൬ݕ௡ାଵ െ2 ݕ ௡൅ݕ ௡ିଵ ∆ଶ൰൅݌ ௡ሺݐሻ. ሺB. 7ሻ Substituting ݕ௡ାଵ value from qn .24 into 2.26 gives, E 2 Eqn ቆ1 ൅2ܯ ௣ ܣߩ ௡1 ∆൅ܯ௣ ܣߩ ௡ܣ௡′ ܣ௡ቇݕሷ௡ൌ2ܪ ߩ∆ଶሺݕ௡ିଵ െݕ ௡ሻ൅݌ ௡ሺݐሻ. ሺB. 8ሻ The boundary conditions are applied to the EOM by the Eqn.B.8. The ݊ no. of equations of motion at ݊no. of nodes of the elevator can be re presented in a matrix form as, Texas Tech University, Vijaya Kaithi, December 2008. 113 ەۖ۔ۖۓݑଵሷ ݑଶሷ ڭ ڭ ܺ ௡ݕሷ௡ۙۖۘۖۗ ൌ ۉۈۈۈۈۇ ܪ ߩ1 ∆ଶ ۏێێێۍെ2 1 1െ 21 1െ 2 1 ڰ 2െ 2 ےۑۑۑې ൅ܪ ߩ ۏێێێێێێۍܣ1Ԣሺݎሻ ܣ1ሺݎሻ ܣ2Ԣሺݎሻ ܣ2ሺݎሻ1 2∆ ڰ ݊ܣԢሺݎሻ ݊ܣሺݎሻےۑۑۑۑۑۑې ۏێێێۍ01 െ1 0 1 െ1 0 1 ڰ 00 ےۑۑۑې یۋۋۋۋۊ ەۖ۔ۖۓݑଵ ݑଶ ڭ ڭ ݑ௡ۙۖۘۖۗ ൅ ەۖ۔ۖۓ݌ଵሺݐሻ ݌ଶሺݐሻ ڭڭ ݌ ௡ሺݐሻۙۖۘۖۗ ሺB. 9ሻ Where ܺ௡ in the last element of mass matrix (i.e. coefficient matrix of ሼݑ ሷ௜ሽ ) is defined as ܺ ௡ൌ1൅2ܯ ௣ ܣߩ ௡൅ܯ௣ ܣߩ ௡1 ∆ܣ௡′ ܣ௡ ሺB. 10ሻ Eqn 2.28 is in the form of a multi degree of freedom spring- mass system ሾܯሿሼݔሷሽ൅ ሾܭሿሼݔሽൌሼ 0 ሽ . And the [M] and [K] matrices ar e the coefficient matrices of ሼݑ ሷሽ ݀݊ܽ ሼݑሽ respectively. Texas Tech University, Vijaya Kaithi, December 2008. 114 Figure B.1: Descritization of elevat or into n number of elements Texas Tech University, Vijaya Kaithi, December 2008. 115 APPENDIX C NUMERICAL ANALYSIS OF LINE AR EQUATIONS OF MOTION C.1. Modal Analysis Modal analysis is used to transform the e quations of motion from physical coordinate system to the principal coordinate system using the orthogonality properties of the modes of vibration. The advantage of transforming to principal coordinate system is that we can decouple the equations of motion and each of th e decoupled equations can be solved as a system rather than solving n ordina ry differential equa tions [15]. The equation of motion of elevator for the longitudinal motion including the tidal forces can be written from Appendix B as, ሾܯሿሼݑ ሷሽ൅ሾܭሿሼݑሽൌሼ ݌ ௨ሺݐሻሽ. ሺC. 1ሻ ሼ݌௨ሺݐሻሽ is a column vector defined as ሼ݌௨ሺݐሻሽൌሾ ݌ ௨ሺݔଵ,ݐሻ ݌ ௨ሺݔଶ,ݐሻ …….݌ ௨ሺݔ௡,ݐሻ ሿ ் .Where ݌௨ሺݐ,ݔ ሻ is the sum of tidal forces due to moon and sun acting in the axia l direction on the elev ator at a distance ݔ e surface of earth. away from th ሾܯሿ and ሾܭሿ matrices are given by Eqn B.9 and can be written as, Texas Tech University, Vijaya Kaithi, December 2008. 116 ሾܭሿൌെ ۉۈۈۈۈۇ ܧ ߩ1 ∆ଶ ۏێێێۍെ2 1 1െ 21 1െ 2 1 ڰ 2െ 2 ےۑۑۑې ൅ܧ ߩ1 2∆ ۏێێێێێێۍܣ1Ԣሺݎሻ ܣ1ሺݎሻ ܣ2Ԣሺݎሻ ܣ2ሺݎሻ ڰ ݊ܣԢሺݎሻ ݊ܣሺݎሻےۑۑۑۑۑۑې ۏێێێۍ01 െ1 0 1 െ1 0 1 ڰ 00 ےۑۑۑې یۋۋۋۋۊ ሾܯሿൌ൦1 1 ڰ ܺ௡൪, ሺC. 2ሻ Where ܺ௡ is given by Eqn B.10. By transforming ሼݑሺݐሻሽ to modal or principal coordinate system ሼߟሺݐሻሽ in terms of eigen vector matrix ሾܷሿ ሾ ሿ ሼߟሺݐሻሽ ሺC. 3ሻ ሼݑሺݐሻሽൌ ܷ Where ሾܷሿൌሾሼܷଵሽ ሼܷଶሽ …… ሼܷ ௡ሽሿ and ሼܷଵሽ , ሼܷ ଶሽ…..ሼ ܷ ௡ሽ are the eigen vectors corresponding to the first, second ……. nth modes of vibration. After substituting Eqn C.3 in Eqn C.1, Eqn C.1 can be written as, ሾܯሿሾߟܷ ሷ ሺݐሻሽ൅ ሾܭሿሾܷሿ ሼߟሺݐሻሽൌሼ ݌ ௨ሺݐሻሽ ሺC. 4ሻ ሿ ሼ Multiplying Eqn C.4 by ሾܷሿ் gives, Texas Tech University, Vijaya Kaithi, December 2008. ሾܷሿ்ሾܯሿሾܷሿ ሼߟሷሺݐሻሽ൅ ሾ ܷሿ்ሾܭሿሾܷሿ ሼߟሺݐሻሽൌሾ ܷ ሿ்ሼ݌௨ሺݐሻሽ. ሺC. 5ሻ 117 From modal orthogonality and normaliz ation properties we can write, ்ܷܫሿ ሺC. 6ሻ ሾ ሿ ሾܯሿሾܷሿൌሾ ்ܭ ሾܷሿ ሾሿሾܷሿൌሾ ܮ ሿ ሾ்ܷ݌௨ሺݐሻሽൌሼ ܳ ሺݐሻሽ ሿሼ Where ሾܫሿ is ݊ ൈ ݊ identity matrix, ሾܮሿ is a diagonal matrix with the diagonal elements as the square of natural frequencies of n no. of modes of vibration and ሼܳሺݐሻሽ is the tidal force vector. Using Eqn C.6, Eqn C.5 is given by, ሼߟሷሺݐሻሽ൅ ሾܮሿ ሼߟሺݐሻሽൌሼܳሺݐሻሽ. ሺC. 7ሻ Let’s consider the Eqn C.7 at i node and is given by, ߟపሷሺݐሻ൅߱௜ଶ ߟ௜ሺݐሻൌܳ ௜ሺݐሻ . ሺth C. 8ሻ ܳ௜ሺݐሻ in the above equation is the sum of tidal forces due to moon and sun i.e. ܳ௜ሺݐሻൌ ܳ௠௜ሺݐሻ൅ܳ ௦௜ሺݐሻ. ܳ௜ሺݐሻ can be expanded in terms of fourier series as ܳ ௜ሺݐሻൌܣ ଴൅෍ܣ ௝∞ ୀଵcos൫߱ ௠௝ݐ൯ ൅ ܤ ଴൅෍ܤ ௝∞ ௝ୀcos൫߱ ௦௝ݐ൯, ሺC. 9ሻ ௝ ଵ Where ߱௠௝ൌ߱כ݆ ௠ , ߱ ௦௝ൌ߱כ݆ ௦ , ߱ ௠ൌଶగ ଶହൈଷ଺଴଴ ߱ ݀݊ܽ ௦ൌଶగ ଶସൈଷ଺଴଴ . The Fourier coefficients in Eqn. C.9 are given by, ܣ ଴ൌܳ׬௠௜ሺݐሻ௧ ଴ ݐ݀ ݐܣ, ௝ൌܳ׬௠௜ሺݐሻ௧ ଴ݏ݋ܿ൫߱ ௠௝ݐ൯ ݐ݀ ݏ݋ܿ׬൫߱ ௠௝ݐ൯ଶ ௧ ଴ ݐ݀ , ሺC. 10ሻ Texas Tech University, Vijaya aithi ܤ଴ൌܳ׬௦௜ሺݐሻ௧ ଴ ݐ݀ ݐ K , December 2008. 118 , ܤ ௝ൌܳ׬௦௜ሺݐሻ௧ ଴ݏ݋ܿ൫߱ ௦௝ݐ൯ ݐ݀ ൫ ݐ൯ଶ ௧ ߱ݏ݋ܿ׬ݐ݀ ௦௝ ଴ . The solution of Eqn. C.8 is given by, ߟ௜ሺݐሻൌߟ ௛௜ሺݐሻ൅ߟ ௣௜ሺݐሻ. The particular solution (ߟ௣௜ሺݐሻ ) and homogeneous solutions (ߟ ௛௜ሺݐሻ ) are given by, ߟ ௣௜ሺݐሻൌܣ଴ ߱௜ଶ൅෍ܣ௝ ߱௜ଶെ߱௠௝ଶ∞ ௝ୀଵݏ݋ܿ൫߱ ௠௝ݐ൯ ൅ܤ଴ ߱௜ଶ൅෍ܤ௝ ߱௜ଶെ߱௦௝ଶ∞ ௝ୀଵݏ݋ܿ൫߱ ௦௝ݐ൯, ሺC. 11ሻ ߟ ௛௜ሺݐሻൌെ ቌܣ଴ ߱௜ଶ൅෍ܣ௝ ߱௜ଶെ߱௠௝ଶ∞ ௝ୀଵ൅ܤ଴ ߱௜ଶ൅෍ܤ௝ ߱௜ଶെ߱௦௝ଶ∞ ቍcosሺ߱௜ݐሻ . ௝ୀଵ The total response of the elevator is ݑሺݐሻ ൌ ሾܷሿ ߟሺݐሻ. Texas Tech University, Vijaya Kaithi, December 2008. 119 APPENDIX D NUMERICAL ANALYSIS OF NONLI NEAR EQUATIONS OF MOTION Based on the nonlinear coupled equations of motion derived in Chapter 3, the equations of motion of the ith node are given by ݑ ሷ௜൅2 Ω ݓ ሶ௜െሺݎ଴൅ܮ൅ݑ ௜ሻߗଶെܧ ߩቌߝ௜ᇱሺ1൅ݑ௜ᇱሻ൅ߝ ௜൭ݑ௜ᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻሺ1൅ݑ௜ᇱሻ൱ቍ ൅ݎ݃଴ଶሺݎ଴൅ܮ൅ݑ ௜ሻ ݎଷെ ܲ௨ ܣߩሺܺሻൌ 0 ሺ D. 1ሻ ݒሷ௜െܧ ߩ൭ߝ௜ᇱݒ௜ᇱ൅ߝ ௜ቆݑ௜ᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻݒ௜ᇱቇ൱൅ݎ݃଴ଶݒ௜ ݎଷെ ܲ௩ ܣߩሺܺሻൌ 0 ሺD. 2ሻ ݓ ሷ௜െ2 ݑ ሶ௜ߗെߗଶݓ௜െܧ ߩ൭ߝேᇱݓ௜ᇱ൅ߝ ௜ቆݓ௜ᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻݓ௜ᇱቇ൱൅ݎ݃଴ଶݓ௜ ݎଷെ ܲ௩ ܣߩሺܺሻ ൌ 0 ሺ.ܦ 3ሻ which is valid for i=2 to n-1. The ati rivative s are replaced by sp al de ݕ௜ᇱ ൌݕ௜ାଵെݕ ௜ିଵ 2∆ , ሺD. 4ሻ ݕᇱᇱൌݕ௜ାଵെ2 ݕ ௜൅ݕ ௜ିଵ ∆ଶ ௜ . where ݕcan be replaced by ݒ,ݑ or ݓ depending on the direction of motion under consideration. For i=1, we use ݑ଴ = ݒ ଴ = 0 to solve the above equations of motion. For i= n, we need to make use of the boundary conditions at the point mass. If the elevator is divided in to N number of nodes, the boundary conditions at the end of the elevator are given by Texas Tech University, Vijaya Kaithi, December 2008. 120 ቈܯ௣ሺݑ ሷே൅2 Ω ݓ ሶேെሺݎ଴൅ܮ൅ݑ ேሻߗଶሻ൅ܣܧ ሺܮሻߝேሺ1൅ݑᇱ ேሻ ൅ܯ ௣ݎ݃଴ଶሺݎ଴൅ܮ൅ݑ ேሻ ݎ௣ଷ቉ቤ ௅,௧ൌ 0 ሺD. 5ሻ ቈܯ௣ݒሷே൅ܣܧ ሺܮሻߝேݒԢே൅ܯ ௣ݎ݃଴ଶݒே ݎ௣ଷ቉ቤൌ 0 ሺD. 6ሻ ௅,௧ ܯ ௣ሺݓ ሷேെ2 Ω ݑ ሶேെߗଶݓேሻ൅ܣܧ ሺܮሻߝேݓԢே൅ܯ ௣ݎ݃଴ଶݓே ݎ௣ଷቤ ௅,௧ൌ 0 ሺD. 7ሻ Eqns. E.5, E.6 and E.7 will be solved to find ߝே, which is a function of ݑሷே , ݒሷே , ݓሷே. From Finite difference ulae, we can write form ݑேᇱᇱൌ െ2ݑ ே൅2 ݑ ேିଵ ∆ଶ൅2 ∆ ݑேᇱ ሺD. 8ሻ ݒேᇱᇱൌ െ2ݒ ே൅2 ݒ ேିଵ ଶ∆൅2 ∆ ݒேᇱ ሺD. 9ሻ ݓேᇱᇱൌ െ2ݓ ே൅2 ݓ ேିଵ ∆ଶ൅2 ∆ ݓேᇱ ሺD. 10ሻ Since ݑேᇱ , ݒேᇱ,ݓேᇱ are functions of ݑሷே , ݒሷே , ݓሷே respectively, ݑேᇱᇱ , ݒேᇱᇱ,ݓேᇱᇱ are also functions of ݑሷே , ݒሷே , ݓሷே. ߝே can be writ n as, ߝேᇱൌݑேᇱᇱ൅ݑேᇱݑேᇱᇱ൅ݒேᇱݒேᇱᇱ൅ݓேᇱݓேᇱᇱ ሺD. 11ሻ ᇱte Equations of motion at the Nth can be expressed in terms of three functions ܩ,ܨ and ܪ as, Texas Tech University, Vijaya Kaithi, December 2008. 121 ܨൌݑ ሷே൅2 Ω ݓ ሶேെሺݎ଴൅ܮ൅ݑ ேሻߗଶെܧ ߩቌߝேᇱሺ1൅ݑேᇱሻ൅ߝ ே൭ݑேᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻሺ1൅ݑேᇱሻ൱ቍ ൅ݎ݃଴ଶሺݎ଴൅ܮ൅ݑ ேሻ ݎଷെ ܲ௨ ܣߩሺܺሻൌ 0 ሺD. 12ሻ ܩൌ ݒ ሷேെܧ ߩ൭ߝேᇱݒேᇱ൅ߝ ேቆݑேᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻݒேᇱቇ൱൅ݎ݃଴ଶݒே ݎଷെ ܲ௩ ܣߩሺܺሻൌ 0 ሺD. 13ሻ ܪൌݓ ሷேെ2 ݑ ሶேߗെߗଶݓேെܧ ߩ൭ߝேᇱݓேᇱ൅ߝ ேቆݓேᇱᇱ൅ܣᇱሺܮሻ ܣሺܮሻݓேᇱቇ൱൅ݎ݃଴ଶݓே ݎଷെ ܲ௩ ܣߩሺܺሻ ൌ 0 ሺD. 14ሻ Newton-Raphson method is used to solv e the non-linear equa tions E.12, E.13 and E.14 for the accelerations ݑሷ,ݒሷ and ݓ ሷ as ൝ ݑሷே ݒሷே ݓሷேൡ ௜ାଵൌ൝ ݑሷே ݒሷே ݓሷேൡ ௜െ ۏێێێێێۍ݂߲ ݑ߲ ሷே݂߲ ݒ߲ ሷே݂߲ ݓ߲ ሷே ߲݃ ݑ߲ ሷே߲݃ ݒ߲ ሷே߲݃ ݓ߲ ሷே ݄߲ ݑ߲ ሷே݄߲ ݒ߲ ሷே݄߲ ݓ߲ ሷேےۑۑۑۑۑې ௜ିଵ ቐ݂ሺݑ ሷேሻ ݃ሺݒሷேሻ ݄ሺݓ ሷேሻቑ ௜ ሺD. 15ሻ Displacements of the elevator in axial, meridional and equatorial directions ݒ,ݑ and ݓ are calculated numerically from the acceleration terms. 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