Characterisation and Applications of Aerodynamic Torques on Satellites

215 Pages·2017·9.58 MB·English

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