CHARACTERISATION AND APPLICATIONS OF AERODYNAMIC TORQUES ON SATELLITES A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2017 David Mostaza-Prieto School of Mechanical, Aerospace and Civil Engineering Contents List of Tables 6 List of Figures 7 Nomenclature 15 Acronyms 18 Abstract 19 Declaration 20 Copyright Statement 21 Acknowledgements 22 Dedication 23 1 Introduction 24 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Scope and research objectives . . . . . . . . . . . . . . . . . . . . . . . 28 1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 Spacecraft Drag Modelling 35 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Characterising the environment . . . . . . . . . . . . . . . . . . . . . . 37 2.2.1 Free molecular flow . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 2.3 Interaction between the body and the flow . . . . . . . . . . . . . . . . 44 2.3.1 Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.2 Schamberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.3 Schaaf and Chambre model . . . . . . . . . . . . . . . . . . . . 49 2.3.4 Angular Distribution and Accommodation Coefficient in Space Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4 Solving the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.1 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Numeric modelling of free molecular flow interactions 60 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2 Calculation of free molecular flow aerodynamics . . . . . . . . . . . . . 62 3.2.1 Free molecular flow models . . . . . . . . . . . . . . . . . . . . . 65 3.2.2 Calculation of equation parameters . . . . . . . . . . . . . . . . 74 3.2.3 Shadow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.1 Outputs and post-processing . . . . . . . . . . . . . . . . . . . . 79 3.3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.3 Study case: QB50 free molecular flow aerodynamic database . . 82 4 Methodology to analyse attitude stability of satellites subjected to aerodynamic torques 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Rotational equations of motion . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.1 Pitch stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.2 Roll-yaw stability . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4 Stability derivatives in free molecular flow . . . . . . . . . . . . . . . . 104 4.5 Time domain response . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.5.1 Pitch axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5.2 Roll-yaw axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.6 Stability with varying dynamic pressure . . . . . . . . . . . . . . . . . . 115 3 5 Momentum dumping by means of aerodynamic torques 122 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2 Momentum exchange capacity of the atmosphere . . . . . . . . . . . . . 123 5.3 Dumping momentum using aerodynamic torques . . . . . . . . . . . . . 127 5.3.1 Body “fixed” solar arrays . . . . . . . . . . . . . . . . . . . . . . 130 5.3.2 Coupling with pitch axis and momentum dumping . . . . . . . . 139 5.3.3 Sun-tracking solar arrays . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Associated drag increment . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4.1 Body “fixed” solar arrays associated drag . . . . . . . . . . . . . 147 5.4.2 Sun-tracking solar arrays associated drag . . . . . . . . . . . . . 150 6 Perigee attitude manoeuvres of geostationary satellites during elec- tric orbit raising 153 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.2.1 Environmental torques . . . . . . . . . . . . . . . . . . . . . . . 158 6.3 Optimal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.4.1 Fixed array rotation . . . . . . . . . . . . . . . . . . . . . . . . 169 6.4.2 Optimum solar array rotation . . . . . . . . . . . . . . . . . . . 169 7 Conclusions 176 7.0.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.0.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.0.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.0.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.0.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Bibliography 183 A Numeric Drag Modelling: supplementary material 200 A.1 Barycenter coordinates and point inside a triangle . . . . . . . . . . . . 200 A.2 Shadow analysis algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 204 4 B Rotational equations of motion 208 B.1 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 B.1.1 Inertial Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 B.1.2 Orbital Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 B.1.3 Body Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 B.1.4 Wind Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 B.2 Linearised rotational equations of motion . . . . . . . . . . . . . . . . . 210 C Gauss Pseudospectral Method 213 5 List of Tables 3.1 Gas-surface interaction models . . . . . . . . . . . . . . . . . . . . . . . 66 3.2 Variables stored in the individual result file . . . . . . . . . . . . . . . . 79 4.1 Sufficient conditions for stability . . . . . . . . . . . . . . . . . . . . . . 104 4.2 Center of gravity position and estimated C . . . . . . . . . . . . . . 111 mα 5.1 Typical ∆V to maintain altitude . . . . . . . . . . . . . . . . . . . . . 150 6.1 Satellite geometric properties . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Initial orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.3 Satellite actuator properties . . . . . . . . . . . . . . . . . . . . . . . . 166 6.4 Boundary values of the problem . . . . . . . . . . . . . . . . . . . . . . 167 6 List of Figures 1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 Variation of the mean atmospheric density with altitude for low, mod- erate and high solar and geomagnetic activities as defined by JB2006 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 VariationoftheEarth’satmosphericcompositionwithaltitudeasdefined by NRLMSISE00 model . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Molecular mean free path (λ) . . . . . . . . . . . . . . . . . . . . . . . 41 2.4 Knudsen number variation with altitude for low, moderate and high solar and geomagnetic activities using NRLMSISE-00, and a characteristic dimension of 1 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Classification of flow regimes using Knudsen number . . . . . . . . . . 42 2.6 Hyperthermal (s → ∞) and hypothermal (s << ∞) flows . . . . . . . . 43 2.7 Impact on parallel surface due to random thermal motion of the flow molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.8 Incident and reflected fluxes on a convex element of area . . . . . . . . 44 2.9 Specular and diffuse reflected fluxes . . . . . . . . . . . . . . . . . . . . 47 2.10 Schamberg’s GSIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.11 Schematic of the main environmental factors affecting accommodation coefficient and angular distribution in low earth orbit . . . . . . . . . . 50 2.12 Uncertainties in drag coefficient caused by quasi-specular remission . . 51 2.13 Schamberg’s quasi-specular and quasi diffuse drag coefficients for simple geometries (based on projected area) . . . . . . . . . . . . . . . . . . . 53 2.14 Comparison of existing computational approaches to spacecraft aerody- namics in low earth orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7 2.15 Surface Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.16 The importance of multiple reflections. If reflected molecules are ignored both surfaces will have the same force acting upon them. In reality, surface B will have less force exerted on it . . . . . . . . . . . . . . . . 59 3.1 Sample of an OBJ file . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 Geometric, body and wind frames for an imported surface mesh . . . . 63 3.3 Two limiting cases, specular and full accommodated diffuse, using the Schaaf and Chambre flat plate model. Equation parameters, s = 7, T = 300K, T = 1000K . . . . . . . . . . . . . . . . . . . . . . . . . 67 w inf 3.4 Sentman diffuse flat plate model for different accommodation coefficients, α . Equation parameters, s = 8, T = 300K, T = 1000K . . . . . . 70 acc w inf 3.5 Sentman diffuse flat plate model for different speed ratios. Equation parameters, α = 0 8, T = 300K, T = 1000K . . . . . . . . . . . . 71 acc w inf 3.6 Comparison between Sentman and Cook models for low (left) and high (right) speed ratios. Equation parameters, α = 0 8, T = 300K, acc w T = 1000K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 inf 3.7 Schamberg quasi-specular model. For values of ν = 1 and α = 0 the acc not accommodated specular model of Schaaf and Chambre is obtained. Goodman curves uses ν = 2 212, φ = 25 degrees and the Goodman 0 model for α with µ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 73 acc 3.8 Temperature and mean molecular mass for low, mean and high solar activities using NRLMSISE-00 . . . . . . . . . . . . . . . . . . . . . . . 74 3.9 Co-rotating speed ratio assuming a circular orbit for low, mean and high solar activities using NRLMSISE-00 . . . . . . . . . . . . . . . . . . . . 76 3.10 Shadow analysis. Forward-facing panes G and back-facing panels G . 77 1 2 3.11 Change of sign in triangles containing the origin. Triangle 1 has two sign changes in z and y. Triangle 2 has only one in z, so it does not contain the origin. Triangle 3 has also two changes, but does not contain the origin; the condition is necessary but not sufficient. . . . . . . . . . . . 78 3.12 Shadow patterns at different angles of attack . . . . . . . . . . . . . . . 78 3.13 Plots of pressure (left) and shear stress (right) surface distributions . . 80 8 3.14 Drag coefficient for a sphere, the numeric solution converges towards the exact solution as the order of the surface mesh increases . . . . . . 81 3.15 Drag de-orbit device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.16 DDD: Force coefficient in x and z-body direction . . . . . . . . . . . . . 82 3.17 Surface mesh representing the cubesat geometry for FMF calculations . 83 3.18 Variation of QB50 drag and lift coefficients with angle of attack and sideslip. Sentman model with reference parameters, α = 0 8, s = 8, acc T = 300K and T = 1000K . . . . . . . . . . . . . . . . . . . . . . . 84 w inf 3.19 Variation of QB50 pitch moment coefficient with angle of attack and sideslip. Sentman model with reference parameters, α = 0 8, s = 8, acc T = 300K and T = 1000K . . . . . . . . . . . . . . . . . . . . . . . 84 w inf 3.20 Effect of shadowed regions in drag (a-c) and moment (b-d) coefficients. For small β the contributions of shadowed parts remains small (top), as β increases the shadowed regions also increase. . . . . . . . . . . . . . . 86 3.21 Pressure distribution for α = 40 deg and β = 20 deg. Shadowed regions appear as areas with c = 0 . . . . . . . . . . . . . . . . . . . . . . . . 87 p 3.22 Evolution of drag coefficient at zero angles of incidence with altitude and speed ratio for low, mean and high solar activities using NRLMSISE-00 87 3.23 Evolution of pitch moment coefficient at α = 10 deg with altitude and speed ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.24 Evolution of drag and lift coefficients at zero angles of incidence with different positions of the right (δ ) and left (δ ) flaps . . . . . . . . . . 89 1 2 3.25 Evolution of roll, pitch and yaw moment coefficients at zero angles of incidence with different positions of the right (δ ) and left (δ ) flaps . . 89 1 2 4.1 Variation of C and C with angle of attack (α) and sideslip (β). Co- m l efficients calculated with ADBSat using Sentman’s model, S = ref 0 12857m2, L = 0 28m. . . . . . . . . . . . . . . . . . . . . . . . . . 96 ref 4.2 Variation of C and C with sideslip (β) and angle of attack (α). Co- l n efficients calculated with ADBSat using Sentman’s model, S = ref 0 12857m2, L = 0 28m. . . . . . . . . . . . . . . . . . . . . . . . . . 97 ref 4.3 For small angles, relationship between angle of attack (α), pitch (θ) and δ (left) and sideslip (β), yaw (ψ) and δ (right) . . . . . . . . . . . . . 97 1 2 9 4.4 Conditions I, II and III (Eqs. 4.35 to 4.37) for U C = 0. The regions 1 n β where the values of σ and σ satisfy the three of them are represented x z in grey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Variation of the conditions I, II and III (Eqs. 4.35 to 4.37) for positive and and negative values of C (bold lines correspond to U C = 0). n 1 n β β Notice that the vertical boundary of condition II (σ = 0) is not affected. 101 x 4.6 Variation of the the stability boundary of the pitch condition in the σ −σ plane for positive and negative values of C . The grey area x z mα represents those values of σ and σ compliant with the inequality for z x C = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 mα 4.7 Roll-yaw stability regions in the σ −σ plane for different values of the z x stability derivatives C and C . a) Classic GG problem with 0 value mα nβ of the aerodynamic coefficients, b) Total region of stability increases for C < 0 and C > 0, c) Total region of stability decreases for C > 0 mα nβ mα and C < 0 d) stability region when the three sufficient conditions are n β met . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.8 FMF Static pitching coefficient for a cylinder (L/D=1) referred to the base area and diameter, for various positions of the centre of mass. . . . 106 4.9 FMF Static pitching coefficient for a cylinder (L/D=5) referred to the base area and diameter, for various positions of the centre of mass. . . . 106 4.10 FMF Static pitching coefficient for a cone (half angle = 18.5 deg) referred to the base area and diameter, for various positions of the center of mass.107 4.11 FMF Static pitching coefficient for a cone (half angle = 45 deg) referred to the base area and diameter, for various positions of the centre of mass.107 4.12 Cylinder (L/D = 5). Variation of longitudinal stability derivative (C ) mα with center of gravity position for different accommodation coefficients (right) and surface temperatures (left). . . . . . . . . . . . . . . . . . . 108 4.13 Cone (half angle = 45 deg). Variation of longitudinal stability deriva- tive (C ) with center of gravity position for different accommodation mα coefficients (right) and surface temperatures (left). . . . . . . . . . . . . 109 10