AUSAT Final Report

8/10/2019 AUSAT Final Report

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H STATIC LOADING ANALYSIS

Figure 96: Compressive Force in Panel Divided by Critical Compressive Force

Acceleration, a

Density,

Cross section area, A

Length of components, L

The sensitivity in the calculated shear stress, , to a valriable, y, is given by Eq. 139:

y

a2 L1+

A2 A1 L2+

A3 A1

L3

y(139)

y = a, , A1, A2, L1, L2

Subscripts refer to different sections of the structure

This sensitivity analysis is a general analysis for all of the stress calculations to gain an

understanding of the effects of changes in the governing variables. General values for

each of the variables will therefore be used, these are listed in Table 34:

Sensitivity to Acceleratin Let y = acceleration, a, Eq. 139 becomes Eq. 140:

a

a2 L1+

A2 A1

L2+ A3 A1

L3

a=

2L1 +

A2 A1

L2 + A3 A1

L3 (140)

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Table 34: Sensitivity Analysis ValuesVariable Value

Acceleration, a [m/ s2] 100Density, [kg/ m3] 2730

Cross Section Area of Stressed Section, A1 [m2] 5105

Cross Section Area of Section Effecting Stress A2, A3 [m2] 510

5

Length of Stressed Section, L1 [m] 0.1Length of Section Effecting Stress, L2, L3 [m] 0.1

Eq. 140 shows that if the value of acceleration increases by 10%, the shear stress in

the section increases by approximately4095 N / m2. This increase is 8 104% of the yield

stress.

Sensitivity to Density Let y = density, , Eq. 139 becomes Eq. 141:

a2 L1+

A2 A1

L2+ A3 A1

L3

= a

2L1 +

A2 A1

L2 + A3 A1

L3 (141)

Eq. 141shows that if the density of the material increases by 10%, the shear stress in

the section increases by approximately 8190 N / m2. This increase is 1.6 103% of the yield

stress.

Sensitivity to Cross Section Area of Stresses Section Let y = area of stressed section,

A1, Eq. 139 becomes Eq. 142:

A1

a2 L1+

A2 A1

L2+ A3 A1

L3

A1= a

2 A2 A21

L2 + A3 A21

L3 (142)

Eq. 142 shows that if the cross section area of the stressed section descreases by 10%,

the shear stress in the section increases by approximately 2730 N / m2. This increase is

5.4104% of the yield stress.

Sensitivity to Cross Section Area of a Section Effecting the Stressed Section Let y =

area of section effecting stress, A2, Eq. 139 becomes Eq. 143:

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A2

a2 L1+

A2 A1

L2+ A3 A1

L3

A2= a

2L2

A1(143)

Eq. 143 shows that if the cross section area of a section effecting the stressed

section increases by 10%, the shear stress in the section increases by approximately

1.83104N / m2.

Sensitivity to Length of Stresses Section Let y = length of stressed section, L1, Eq. 139

becomes Eq. 144:

L1

a2 L1+

A2 A1

L2+ A3 A1

L3

L1= a

2 (144)

Eq. 144 shows that if the length of the stressed section increases by 10%, the shearstress in the section increases by approximately 1365 N / m2. This increase is 2.710

4% of

the yield stress.

Sensitivity to Length of Section Effecvting Stressed Section Let y = length of stress

section, L2, Eq. 139 becomes Eq. 145:

L1

a2 L1+

A2 A

1L2+

A3 A

1L3

L2 = a

2 A

2 A1 (145)

Eq. 145 shows that if the length of a section effecting the stressed section increases by

10%, the shear stress in the section increases by approximately 1365 N / m2.

Discussion The sensitivity analysis shows that a ten percent change in a governing

variable will have a maximum effect of increasing the the shear stress by 1.6 103% of

the yield stress. This sensitivity analysis justies the simplications made to the model.

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I RANDOM VIBRATION FEA ANALYSIS REPORT

I Random Vibration FEA Analysis Report

A random vibration analysis of the structure was conducted in ANSYS, the report for this

analysis is as follows:

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As part of the final year project Design, build and launch of a small satellite based on CubeSat

designs random vibration analysis was performed for launch vehicle integration qualification. The

project is being undertaken by five undergraduate students and is called AUSAT. The following

analysis was a preliminary validation to determine whether the satellite structure could withstand

loading due to launch vehicle vibrations. This directly related to the CubeSat standards which statethat to prove flightworthiness random vibration testing must be completed at a level higher than the

published launch vehicle envelope outlined in the Mission Test Plan (MTP).

The finite element analysis (FEA) package, ANSYS Workbench, was used to simulate random

vibrations present during the launch of the satellite. The model of the satellite was constructed in Pro

Engineer Wildfire in sufficient detail to be unambiguously constructed by an external workshop at BAE

Systems. The model was then defeatured to improve mesh quality without significantly altering the

design. Firstly a static structural pre-loading was applied to the internal rails to simulate the loads of

internal electronics. A modal analysis was conducted then conducted in order to determine the natural

frequencies of the satellite. Following the modal analysis, stochastic vibration loads were applied to

the structure to determine the maximum stresses and deformation of vital components.

Flight worthiness of the satellite will be granted if the maximum Von-Mises stresses of the structure

are below the yield stress and also the maximum deflection of the solar panel printed circuit boards

(PCBs) are within specified limits. Success in these two criteria will mean that the structure will not fail

and the solar cells will not break during launch. Although FEA alone is not sufficient to evaluate

launch qualification it provides a preliminary check of any major problems before a prototype is

constructed. Experimental results obtained later in the year will validate launch qualification and FEA

analysis.


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In this analysis it was essential to model the entire satellite despite the satellite having an axis of

symmetry. This was due to the asymmetric mode shapes in the modal response. Therefore, modelling

half the satellite and applying symmetric boundary conditions would not accurately model the modal

response of the structure. The model was adapted from a computer aided design (CAD) model, Fig. 1,

which was constructed in Pro Engineer for the final year project, AUSAT. The model was adapted bydefeaturing the model to improve the quality of the mesh. The CubeSat model was simplified in a

number of areas for the ANSYS analysis in order to make the system solvable. The screws and screw

holes were removed to improve the mesh on the side panels. This simplification has also been used to

appropriately perform a random vibration analysis of a similar pico-satellite structure (Pierlot, 2009). As

electronic components have not been finalised in the final year project a detailed model of the boards

were omitted. However, the approximate mass of the boards were know allowing a load to be applied

to the internal rails to simulate the boards. The pre-loading applied represented a static loading of the

mass of the board at 10g. Also all roundings on rails and cross brackets were removed to increase

mesh quality and reduce the number of small angled elements.

Figure 1: Pro Engineer Wildfire detailed CAD model


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A static structural pre-loading, representing internal electronic boards, and two types of analysis were

required to determine the launch qualification of the satellite. A modal analysis was conducted in order

to find the natural frequencies of the satellite structure. The launch vehicle standards require any

resonance frequencies below 2000 Hz to be analysed by a random vibration test. The random

vibration analysis that was conducted used a power spectral density (PSD) specified by testing

requirements for the Dnepr launch vehicle, Appendix C.

The mesh was defined in the static pre-loading and verified in the modal analysis before determining

the natural frequencies. As the geometry was imported from a 3D CAD model the automatic element

type where chosen as 3D tetrahedral. Firstly an automatic mesh was applied to the model, Fig. 2.

This mesh size obtained accurate results in the verification model and would be sufficient to model

areas of uniform geometry where detailed solution are not required such as the side panels.

Figure 2: Initial automatic tetrahedral mesh applied to entire structure

The frame structure and cross brackets were not crucial areas of analysis, which was seen in

preliminary random vibration analysis, and therefore the mesh size of these components were left

coarse. The meshes of these components were mapped reducing irregular shaped elements and

improving the quality of the mesh, seen in Fig. 3.


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Figure 3: Manual meshing of the frame and cross brackets to reduce irregular shaped elements.

The PCBs, which supported the solar cells, were refined to a 5 mm element size, Fig. 4. This was

done after preliminary modal and vibration analysis which indicated that the mode shapes and

vibrations affected the top and bottom panel more so than the structural panels, frame and cross

brackets. Also this was necessary to determine accurate deflections of the PCBs to evaluate whether

the solar cells would be damaged, deflection under 1mm.

Figure 4: Refinement of mesh of PCB to 5 mm.


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Determination of the damping coefficient was important to model the satellite appropriately. A similar

CubeSat project, OUFTI, analysed random vibrations for launch qualification. The quality factor

(amplification factor) of the OUFTI CubeSat structure was estimated at Q = 10 (Galli, 2008). Using the

following relationship (Roberts, 2009),

the damping coefficient of the satellite could be found by reaaranging to the following,

The frame, structural panels and cross brackets of the satellite are made from aluminium 6061-T6.

This material was chosen for the primary structure as it is lightweight and recommended by CDS

(Munakata, 2008). The PCBs were modelled as RF-4 which is a common material used in electronicboards (Orly, 2009). The spacers between PCBs and structural panels were modelled as

Polytetrafluoroethylene (PTFE) to reduce wear and provide appropriate support to the PCBs. The

internal rails, used to support electronics, were modelled as structural steel as these were vital in

distributing the load of electronics and important to have higher strength than aluminium 6061-T6. The

materials properties used in the analysis can be seen in Table 3.

Table 3: Material properties used in analysis of CubeSat

Material

Properties

Youngs

Modulus (GPa)

Density

(kg/m3

)

Poissons

Ratio

Tensile yield

Strength (MPa)

Coefficient of thermal

expansion (10-5

x C-1

) Aluminium 6061-T6 68.9 2700 0.33 276 2.4

RF-4 18.6 1820 0.136 276 1.2

PTFE 1 2200 0.46 20 13

Structural steel 200 7850 0.3 250 1.2

The boundary conditions change for each direction analysed as per the testing requirements for the

Dnepr Launch Vehicle. The testing requirements state that a CubeSat is to be oriented on a shaker in

the x, y and z directions and a vibration analysis performed for each of the three axes. For the x and y

directions the fixed support is located on the surfaces of the rails perpendicular to the axis direction.

For the z direction the fixed support is placed on the bottom of the four rails. In random vibration

analysis all supports that are not fixed are automatically assigned as free boundaries, which would be

the case in determining random vibrations in each axis individually.


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First a static structural preloading was applied to the internal rails of the satellite to represent the

weight of electronics, Fig 5. The magnitude of this load was determined by multiplying the mass of the

internal electronics, 500 grams, by a constant acceleration of 10g which is present in the launch of the

Dnepr launch vehicles. The load was applied at the central locations of the rail as exact location of

electronic boards have not been finalised. This would over estimate the stresses in the rails as the

bending moment would be maximised in this instance.

Figure 5: Representation of the loads of electronic boards on internal rails

A number of satellite launch providers offer CubeSat launches, as a secondary payload, to Low Earth

Orbit (LEO) however as a launch has not yet been finalised for this analysis the Dnepr launch vehiclewill be selected to provide random vibration statistics. The Dnepr launch vehicle is a Russian rocket

that has successfully launched seven CubeSats and had one failure during launch, destroying 14

CubeSats. The Dnepr launch vehicle currently has the most severe vibration response and it is likely

if the CubeSat structure can withstand a Dnepr launch it will be qualified for all launch vehicles

(CubeSat, 2009). The spectral density for each frequency range for both the high level and low level

qualification profile for a typical Dnepr Launch are shown in Table 4 and Table 5 respectively. The

high level qualification profile must be applied to the CubeSat for 35 seconds and the low level

qualification profile for 831 seconds to simulate a typical launch. These loads will be applied through

the fixed support boundary conditions, simulating the physical connection of the CubeSat and

experimental shaker.


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A verification model was first constructed to determine an appropriate mesh size for the CubeSat finite

element model (FEM). Once a converged solution was determine for the verification model by

comparing analytical solutions to FEA the CubeSat FEM was analysed. First a static structural pre-

loading was applied to simulate the electronic boards and then modal response of the satellite

structure was determined and all natural frequencies below 2000 Hz were tabulated. Once the modalanalysis was complete a random vibration analysis was performed for each of the CubeSat primary

axis. Von Mises stress and deformation in each of the CubeSat axis was investigated to determine

whether the qualification parameters, Table 2, were satisfied.

In order to validate the fidelity of random vibration analysis of the CubeSat model two verification

models were constructed. The two verification models were of a cantilever beam and a flat plate

clamped at one edge. These simplified models were chosen as the CubeSat structure is a

combination of both beam and plate components. Additionally, analytical solutions have been welldocumented to provide a reliable validation method. Hand calculations, Appendix A, were performed

for each of the models to determine the natural frequency using analytical methods. A modal analysis

was then performed using ANSYS Workbench on both models. The geometries of the models were

constructed in the design modeller section and aluminium alloy properties were assigned in the

material library. A detailed material model was not developed in the verification stage as the

aluminium alloy in the general material library was sufficient to validate convergence. The mesh sizes

of both models were varied to determine when the solution had converged. Varying the mesh size also

allowed for the number of nodes and elements to be recorded which would affect the size of the mesh

that could be used in the CubeSat FEM. The following sections detail the verification procedure.


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The first model constructed was a simple cantilever beam, Fig. 6. This beam represents the four rails

of the CubeSat that are constrained within the Poly-Picosatellite Orbital Deployer (P-POD). Hand

calculations were undertaken to determine the fundamental natural frequency of the beam using

theory highlighted in Inham. These analytical solutions were compared to a modal analysis performed

in ANSYS Workbench. Fig. 6 shows the model with a fixed support at the left face to simulate the

cantilever.

Figure 6: Cantilever beam model used in verification, left face is fixed.

A modal analysis was selected in ANSYS Workbench to determine the fundamental natural frequency

using FEA methods. The design modeller was used to first construct a square of width 8.5 mm and

then extruded to form a bar of 100 mm length. Although beam elements would accurately model the

bending modes of a cantilever beam, solid elements was used as it was necessary to model torsional

and lateral modes in the satellite structure. Also, simplifying the CubeSat FEM as beams and plates

removes additional stiffness at thickened joints and therefore would not accurately model the naturalfrequencies of the satellite structure.

To test for convergence and determine an appropriate mesh size for the CubeSat FEM the mesh size

for the beam was set at 15 mm to form a coarse mesh and then decreased to 8 mm to form a fine

mesh, Fig. 7 left and right respectively.

Figure 7: Left: Cantilever with 15 mm mesh size. Right: Cantilever with an 8 mm mesh size.


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The material properties used in both the FEA and hand calculations for the cantilever beam

verification model were generic aluminium alloy, Table 6. This was sufficient as the verification model

was analysing solution convergence of an implicit material not the dependence of material properties

on the solution.

Table 6: Material properties of aluminium alloy used inverification modelProperty Value

Youngs Modulus 71.0 GPa

Density 2770 kg/m

Poissons Ratio 0.33

A fixed support was applied to one of the end surfaces to form the cantilever, Fig. 6.

The result of the analytical method is compared to the FEA model of the cantilever beam for a variety

of meshes, Table 7. The percent error from the analytical fundamental frequency was found for mesh

sizes of 15, 10, 9 and 8 mm. It can be seen that there is a significant difference in frequencies

between 8 and 9 mm. This is because the element size is larger than the beam width until 8 mm. It

should be aimed to at least have the element size less than the minimum dimension of the beam.

Table 7: Comparison of Analytical and FEA Solutions for Beam Verification Model.


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The second verification model consisted of a flat plate that was clamped at one edge, Fig. 8. The

reason this model was chosen is similar to the cantilever beam, it resembles the panels found on the

satellite and analytical solution have been well documented.

Figure 8: Verification model of a flat plate clamped along left edge.

Similar to the cantilever beam the flat plate was modelled in design modeller by first making a square

plate with length of 100 mm and then extruding by 1.8 mm to form a flat plate. A thickness of 1.8 mm

was chosen in the verification model as this matches the panel thickness used in the CubeSat design.

To test for convergence and determine an appropriate mesh size for the CubeSat FEM the mesh sizefor the beam was set at 15 mm to form a coarse mesh and then decreased to 2.5 mm to form a fine

mesh, Fig. 9 left and right respectively.

Figure 9: Left: Flat plate with 15 mm element size. Right: Flat Plate with a 2.5 mm element size.


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Similar to the beam verification model aluminium alloy was chosen from the general materials material

library in engineering data. The specifications for this material are listed in Table 6.

A fixed support was applied to one of the edge surfaces to form the clamped plate, Fig. 8.

The result of the analytical method is compared to the FEA model of the flat plate for a variety of

meshes, Table 8. The percent error from the analytical fundamental frequency is not applicable as the

analytical solution only estimates an upper bound for the fundamental frequency. The analysis was

performed at a mesh size of 1 mm as well but this exceeded the node limit of the ANSYS licence.

Table 8: Results of Flat Plate Verification Model


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The verification models indicated that there were no appreciable differences in natural frequencies by

decreasing the mesh size below 8 mm and 5 mm for panel and beam respectively. The only major

effect of decreasing mesh size below this size is to increase the node count. Also identified was the

significant decrease in errors by limiting the element size to less than the minimum dimension of

meshed component. Therefore, to comply with these results and avoid maximum node limits the frame

and cross brackets were meshed using 8.5 mm tetrahedral elements. As the deflection of the solar cell

PCBs was required to determine whether the solar cells would be damaged the mesh was refined for

these components to 5 mm. The element sizes used in satellite FEM are summarised in Table 9.

Table 9: Mesh Sizes for CubeSat FEM

Component Mesh Size (mm)

Frame 8.5 mm

Cross Brackets 8.5 mm

Top & Bottom Panel 10 mm

Side Panels 10 mm

Solar cell PCB 5 mm


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Modal and random vibration analysis was performed on the simplified CubeSat FEM constructed

using the appropriate mesh size obtained in the verification model, boundary conditions, damping and

material properties obtained in review of literature highlighted in previous sections. First the modal

response of the satellite was obtained and all natural frequencies below 2000 Hz were tabulated. Von

Mises stress and displacement in each axis were then evaluated using the PSD stated in the CDS for

Dnepr launch vehicles. The stress distribution and maximum deflection of each axis random vibration

analysis were then tabulated to evaluate whether the objectives of the analysis were satisfied.

Modal analysis of the CubeSat structure indicated that there were 40 modes, Fig 10 & Table 10, under

2000 Hz that would be of particular interest in random vibration analysis. There were three distinct

frequencies ranges corresponding to modes 1 to 5, 6 to 20 and 21 to 40 which if prolonged excitation

was presented would lead to large increases of stress and deflections. It would be possible to shift thehigher frequency range, modes 21 to 40, above 2000 Hz through the addition of mass to the structure.

This would be advantageous as the launch vehicle and deployer structure would then damp out the

random vibrations above 2000 Hz reducing the likelihood of excitation of CubeSat natural frequencies.

However, as several components of the satellite have not been finalised it was important to model the

empty structure as a worst case scenario.

Figure 10: The modal frequencies of all modes below 2000 Hz of the CubeSat structure


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Table 10: Modal response of the CubeSat structure

Mode Frequency Mode Frequency Mode Frequency Mode Frequency

1 657.14 11 1107.8 21 1674.3 31 1902.9

2 657.4 12 1116.2 22 1733.2 32 1907.1

3 658.45 13 1162.1 23 1820.9 33 1920.9

4 768.58 14 1162.8 24 1850.5 34 1922.6

5 774.77 15 1163.4 25 1888.4 35 1935.5

6 1080 16 1208.4 26 1891.4 36 1942.9

7 1084.6 17 1209.9 27 1896.7 37 1944.6

8 1084.8 18 1212.4 28 1899 38 1979.5

9 1086.3 19 1319.5 29 1899.3 39 1980.5

10 1091.1 20 1322.2 30 1901 40 1981.9

To model random vibrations in the x-axis fixed boundary conditions, simulating connection between

test pod and structure, were applied to the side of the rails of the CubeSat. The results of the x-axis

random vibration analysis are summarised in Table 11. The Von Mises stress distribution, shown in

Fig 11, was maximum in the solar cell PCB at the connection between the PCB and the spacers.

Illustrations of the maximum deflection which occurred at the centre of the solar cell PCB and can be

seen in Fig 12.

Figure 11: Von Mises stress for x-axis random vibration analysis.

Figure 12: X-axis displacement for x-axis random vibration analysis.


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Similarly y-axis vibrations were modelled with fixed boundary conditions applied to the face of the

frames of the CubeSat. The results of the y-axis random vibration analysis are summarised in Table

11. The Von Mises stress distribution, shown in Fig 13, was maximum in the solar cell PCB at the

connection between the PCB and the spacers. Illustrations of maximum deflection which occurred at

the centre of the solar cell PCB can be seen in Fig 13 and 14 respectively.

Figure 13: Von Mises stress for y-axis random vibration analysis.

Figure 14: Y-axis displacement for y-axis random vibration analysis.

Finally z-axis vibrations were modelled with fixed boundary conditions applied to the bottom face ofthe CubeSat rails. The results of the z-axis random vibration analysis are summarised in Table 11.

The Von Mises stress distribution, shown in Fig 14, was maximum in the solar cell PCB at the

connection between the PCB and the spacers. Illustrations of the maximum deflection which occurred

at the centre of the top solar cell PCB can be seen in Fig 15.


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Figure 15: Von Mises stress for z-axis random vibration analysis.

Figure 16: Z-axis displacement for z-axis random vibration analysis.


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The random vibration analysis indicated that the maximum deflection occurred in the direction of

excitation, as expected. The location of the maximum deflection occurred in the centre of the solar

cell PCB for each loading condition. All deflections were below the specification of maximum

deflection of the solar cell of 1mm, shown in Table 11. The maximum deflection throughout the

analysis was the X axis excitation, with a deflection of 0.02027mm, which corresponds to a factor of

safety of 49. This ensures the solar cells will survive the launch environment and will be operational on

orbit.

As indicated by previous literature, failure of the structure is unlikely during launch. This has been

verified by the stress distribution obtained during the random vibration analysis, results shown in Table

11. The maximum stress was shown to occur in the Y axis excitation, between the solar cell PCB and

the spacer. The value of stress recorded was 1.41MPa, this results in a factor of safety 14 for the RF-

4 PCB material. The stresses in the satellite structure were considered negligible, resulting in a

successful result for random vibration analysis.

Table 11: Results of Random Vibration Analysis of the CubeSat

Direction of Vibration X Deflection

(mm)

Y Deflection

(mm)

Z Deflection

(mm)

Stress

(MPa)

X 0.02027 0.00060 0.00109 1.10

Y 0.00064 0.01980 0.00076 1.41

Z 0.00073 0.00074 0.01692 1.36

During the modelling of the satellite some simplifications that were made would change modal

response. These simplifications include the removal of screw holes and screws, simplifying curves to

rectangular sections and neglecting internal components of the satellite. These simplifications would

change the mass distribution and total mass, resulting in lower natural frequencies. Further analysis is

required upon finalisation of satellite components. Additionally cut outs for wiring on the solar cell

PCB and structural panels were neglected. This would reduce the structural rigidity of the side panels

therefore decreasing the natural frequency. However these simplifications were necessary improve

the mesh quality and satisfy the node limit of 32000 nodes. The mesh body sizing feature in ANSYS

was used after verification modelling to optimise the number of nodes with the quality of the mesh.The resultant number of nodes utilised was 31343.

Shielding of the satellite from nose cone and deployer would maintain the temperature of the satellite

at ambient conditions. Thermal analysis will therefore not be required in this analysis of the launch of

the satellite.


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The random vibration analysis using ANSYS posed some potential problems both in the software itself

and also in the system being analysed. Firstly, the ANSYS licence at The University of Adelaide only

allows for a maximum of 32000 nodes, which restricts the refinement of the mesh and therefore the

potential accuracy of the results. The CubeSat CAD model itself also posed a problem in our analysis

due to its complexity. As a result the model had to be simplified in order to for an appropriate mesh tobe generated.

The results of random vibration analysis at Dnepr launch vehicle levels indicated that the maximum

deflections of the solar cell PCBs were well below the specified limit of 1 mm. Additionally, the stress

within all structural components was significantly below the yield strength of aluminium 6061-T6.

These results would indicate that both the structure and solar cells would not be damaged in the

launch environment allowing successful operation of the satellite once in orbit. To further validate the

results of FEA and satisfy launch qualification regulations random vibration testing will also be

performed on the fully constructed structure, including solar cells and internal electronics.


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J MAGNETORQUER TEST THEORY

J Magnetorquer Test Theory

The torque created by a magnetorquer has to be calculated so that the attitude of the

satellite can be controlled. A test suggested in the thesis Development of an Active

Magnetic Attitude Determination and Control System for Picosatellites on highly inclined

circular Low Earth Orbits by Jens Gieelmann from RMIT will be implemented.

Test Setup

Apparatus:

Helmholtz Coil Pair

Light Triggered Timer

Retort stand

Coil

Schematic - Figure 97:

Figure 97: Schematic of Experiment Setup

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The coil is set up so that it is between the Helmholtz Coil Pair in the uniform magnetic

eld. The plane of the coil is to be perpendicular to the magnetic eld where to torque is

a minimum. The retort stand will be set up to hold up the coil by a wire supplying the

current to the coil. The wire supporting the coil will allow the coil to oscillate to small

angles and the period of oscillation will be calculated using the Light Triggered Timer.

Theory

To measure the torque as a function of theta created by the coil it is necessary to sum the

forces acting on this system. The two main forces acting on the system are the force due

to the current in the coil and the magnetic eld strength and the force due to the torsional

stiffness of wire suspending the coil. The angle of the magnetorquer is measured from

perpendicular to the magnetic eld, Figure 98.

Figure 98: Schematic of the torqu acting on the system

Calculating the torque due to the current in the coil and the magnetic eld strength:

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Substituting Eqs. 149 and 150 into Eq. 151 gives Eq. 152:

= k MBsin () (152)

The torque acting on the system can be written in terms of the mass moment of inertia

and the angular acceleration of the system Eq. 153:

= J = J d2dt2

= J (153)

Where J is the mass moment of inertia of the magnetorquer coil (kgm2)

Solving Eq. 153: Eq. 154 can be created by substituting Eq. 152 into Eq. 153:

J = k MBsin () (154)In this experiment will be kept small, less than 15, and thus the small angle

approximation, Eq. 155:

sin () (155)Using Eq. 155, Eq. 154 becomes Eq. 156:

+ k + MB

J = 0 (156)

Eq. 156 is in the form of a standard differential equation and can be solved letting = Ae t

and solving the characteristic equation.

The characteristic equation of Eq. 156 is given in Eq. 157:

2 + k + MB

J = 0 (157)

Solving Eq. 157 gives Eqs. 158 and 159:

= k + MB J = i k + MB J = i (158)

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= c1cos ( t) + c2sin ( t) (159)

Eq. 159 has the initial conditions listed in Eqs. 160 and 161:

(t = 0) = 0 Initial Angular Displacement (160)

(t = 0) = 0 Initial Angular Velocity (161)From the initial conditions, Eq. 159 becomes Eq. 162:

= 0cos k + MB J t = 0cos 2 T t (162) Where T is the period of oscillation of the system (s), as measured by the light

triggered timer

Calculation of J:

J, the mass moment of inertia of the magnetorquer coil can be calculated by Eq. 163:

J = mcL2

6 (163)

Where mcis the mass (kg) of the magnetorquer coil.

Calculation of K : To calculate the torsional stiffness of the wire, k , the power to the

coils is to be switched off (making M=0), the coil set to oscillate about = 0 (small ) and

T k is to be measured by the light triggered timer. The torsional stiffness of the wire can

then be calculated by Eq. 162, this is shown in Eq. 164:

k = J 4

2

T 2K (164)

Calculation of M:

M = J 4 2

T 2B k B

(165)

The torque of the coil:

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Now that M is known the equation for the torque of the coil can be found by Eq. 166:

c = MB (166)

In the case of the satellite, B is the magnet eld strength of the earth at a given time. B

would have to be known for the control of the satellite.

Error in Measurement

The total error in the calculation of the torque, c, as calculated by Eq. 167:

c c

2= 4

2

M J

T 2B

2 Mc Mc

2+ 2 LL

2+ BB

2 + 2 T T 2

+ k B

2 Mc Mc

2+ 2 L

L

2+ 2

T k

2+ B

B

2+ B

B2

+

2(167)

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K TESTS RESULTS

K Tests Results

The test results that are included in this appendix are for the following tests:

Magnetorquer Tests - K1

Thermal Vacuum Tests - K2

Vibration Tests - K2

K.1 Magnetorquer Tests

The magnetorquer tests were completed for the three magnetorquers. The results of the

tests for each magnetorquer are shown in the following tables.

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K.1 Magnetorquer Tests K TESTS RESULTS

Magnetorquer 1:

Table 35: Physical properties of magnetorquer 1Magnetorquer physical properties

Mass 1 0.0171 kg

Length 1 0.07566 mHeight 1 0.07339 maverage length 0.074525 m

Moment of Inertia, J1 1.58288E-05 kg.m2wire mass 0.002 kg

wire distance 0.04 mTot. Mom of Inertia 1.90288E-05 kg.m2

Mag eld strength, Beta 0.00178 T

Table 36: Measured and calculated values for magnetorquer 11 Helmholtz Coil Pair: B = 17.8 gauss, V = 8.2volts, I = .75Amps

Magnetorquer volatge and currentV 0volts 7.1volts 13.2volts 18.4volts 26volts 32.2voltsI 0A 0.05A 0.1A 0.15A 0.2A 0.23A

Five measurements of ten periods of oscillation [s]1 5.811 5.639 5.487 5.374 5.28 5.222 5.816 5.638 5.478 5.368 5.291 5.233 5.815 5.639 5.479 5.379 5.294 5.2334 5.81 5.639 5.487 5.376 5.287 5.2275 5.812 5.639 5.484 5.376 5.287 5.23

k [N.m] 2.223 E-3Avg. T [s] 0.58128 0.56388 0.5483 0.53746 0.52878 0.5228

M 0 0.078275 0.15477 0.21197 0.26033 0.29506 .[N.m.deg] 0 -1.3932 -2.7550 -3.7732 -4.6340 -5.2521

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K.1 Magnetorquer Tests K TESTS RESULTS

Magnetorquer 2:

Table 37: Physical properties of magnetorquer 2Magnetorquer physical properties

Mass 2 0.0168 kg

Length 2 0.07509 mHeight 2 0.0728 maverage length 0.073945 m

Moment of Inertia, J2 1.531E-05 kg.m2wire mass 0.002 kg

wire distance 0.04 mtot. Mom of Inertia 1.851E-05 kg.m2Mag eld strength 0.00154 T

Table 38: Measured and calculated values for magnetorquer 22 Helmholtz Coil Pair: B = 15.4 gauss, V = 8.2volts, I = .75Amps

Magnetorquer volatge and currentV 0volts 7.8volts 13volts 19.6volts 27.5volts 32.2voltsI 0A 0.05A 0.1A 0.15A 0.2A 0.22A

Five measurements of ten periods of oscillation [s]1 5.835 5.679 5.569 5.466 5.369 5.3762 5.839 5.674 5.564 5.469 5.382 5.3723 5.84 5.674 5.565 5.472 5.38 5.3794 5.841 5.673 5.561 5.464 5.388 5.3795 5.837 5.674 5.565 5.459 5.385 5.391

k [N.m] 0.002143773Avg. T [s] 0.58384 0.56748 0.55648 0.5466 0.53808 0.53794

M 0 0.08142 0.14024 0.19614 0.24683 0.24769 .[N.m.deg] 0 -1.2538 -2.1598 -3.0206 -3.8013 -3.8144

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K.2 Thermal Vacuum Tests K TESTS RESULTS

K.2 Thermal Vacuum Tests

The results for the thermal vacuum tests are shown in the following tables:

Time T Aim T Amb T 1 T 2 T 3 T 4 Readings from Multimeters

min deg C deg C deg C deg C deg C deg C Amb T1 T2 T3 T4

0 21 21.38 21.33 21.09 21.29 21.19 0.21 207.5 0.22 214.3 0.21

1 21.82 22.01 21.32 21.09 21.27 21.19 0.22 207.4 0.22 214.1 0.21

2 22.63 22.93 21.27 21.09 21.24 21.19 0.2278 206.9 0.2160 213.8 0.2140

3 23.45 23.66 21.36 21.19 21.35 21.29 0.2351 207.8 0.2170 214.9 0.2150

4 24.27 24.50 21.25 21.09 20.31 21.29 0.2435 206.7 0.2160 204.5 0.2150

5 25.08 25.36 21.24 21.09 21.27 21.09 0.2521 206.6 0.2160 214.1 0.2130

6 25.90 26.15 21.22 21.19 21.28 21.09 0.2600 206.4 0.2170 214.2 0.2130

7 26.72 26.64 21.34 21.39 21.52 21.29 0.2649 207.6 0.2190 216.6 0.2150

8 27.53 28.77 21.48 21.49 21.63 21.29 0.2862 209.0 0.2200 217.7 0.2150

9 28.35 28.43 21.19 21.29 21.45 21.09 0.2828 206.1 0.2180 215.9 0.2130

10 29.17 29.26 21.25 22.99 22.86 21.89 0.2911 206.7 0.2350 230.0 0.2210

11 29.98 29.99 21.58 21.79 21.76 21.39 0.2984 210.0 0.2230 219.0 0.2160

12 30.80 31.44 21.48 21.79 21.86 21.29 0.3129 209.0 0.2230 220.0 0.2150

13 31.62 31.76 21.56 22.19 22.09 21.69 0.3161 209.8 0.2270 222.3 0.2190

14 32.43 33.00 24.18 23.39 22.44 22.19 0.3285 236.0 0.2390 225.8 0.2240

15 33.25 33.40 21.58 22.19 22.86 21.39 0.3325 210.0 0.2270 230.0 0.2160

16 34.07 34.02 21.45 22.19 22.24 21.29 0.3387 208.7 0.2270 223.8 0.2150

17 34.88 35.45 21.66 22.39 22.46 21.59 0.3530 210.8 0.2290 226.0 0.2180

18 35.70 35.82 21.56 22.49 22.54 21.39 0.3567 209.8 0.2300 226.8 0.2160

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19 36.52 36.83 21.66 22.69 22.74 21.49 0.3668 210.8 0.2320 228.8 0.2170

20 37.33 36.91 21.81 22.99 23.00 21.59 0.3676 212.3 0.2350 231.4 0.2180

21 38.15 38.40 22.16 23.29 23.33 21.89 0.3825 215.8 0.2380 234.7 0.2210

22 38.97 39.02 22.37 23.59 23.86 22.19 0.3887 217.9 0.2410 240.0 0.2240

23 39.78 40.03 22.46 23.59 23.78 22.39 0.3988 218.8 0.2410 239.2 0.2260

24 40.60 41.00 22.31 24.09 23.84 21.99 0.4085 217.3 0.2460 239.8 0.2220

25 41.42 41.34 22.41 24.09 24.25 22.19 0.4119 218.3 0.2460 243.9 0.2240

26 42.23 42.38 22.59 24.49 24.39 22.69 0.4223 220.1 0.2500 245.3 0.2290

27 43.05 42.98 22.78 24.59 24.51 22.49 0.4283 222.0 0.2510 246.5 0.2270

28 43.87 44.07 22.86 24.79 24.75 22.59 0.4392 222.8 0.2530 248.9 0.2280

29 44.68 44.85 23.07 25.09 25.03 22.79 0.4470 224.9 0.2560 251.7 0.2300

30 45.50 45.71 23.26 25.39 25.28 22.99 0.4556 226.8 0.2590 254.2 0.2320

31 46.32 46.50 23.51 25.79 25.76 23.49 0.4635 229.3 0.2630 259.0 0.2370

32 47.13 47.50 23.71 26.19 25.90 23.29 0.4735 231.3 0.2670 260.4 0.2350

33 47.95 48.37 24.58 27.19 26.31 23.69 0.4822 240.0 0.2770 264.5 0.2390

34 48.77 48.95 24.18 27.19 26.60 24.29 0.4880 236.0 0.2770 267.4 0.2450

35 49.58 49.76 24.48 27.29 26.96 24.19 0.4961 239.0 0.2780 271.0 0.2440

36 50.40 51.13 26.58 28.49 28.42 25.29 0.5098 260.0 0.2900 285.6 0.2550

37 51.22 50.39 25.05 28.09 27.64 24.69 0.5024 244.7 0.2860 277.8 0.2490

38 52.03 52.68 25.20 28.29 27.86 24.79 0.5253 246.2 0.2880 280.0 0.2500

39 52.85 53.10 25.49 28.69 28.26 25.09 0.5295 249.1 0.2920 284.0 0.2530

40 53.67 53.74 25.77 29.09 28.56 25.39 0.5359 251.9 0.2960 287.0 0.2560

41 54.48 54.70 26.08 29.49 29.01 25.69 0.5455 255.0 0.3000 291.5 0.2590

42 55.30 55.48 26.44 29.89 29.46 26.09 0.5533 258.6 0.3040 296.0 0.2630

43 56.12 56.31 26.76 30.29 29.86 26.29 0.5616 261.8 0.3080 300.0 0.2650

44 56.93 57.15 27.07 30.69 30.29 26.69 0.5700 264.9 0.3120 304.3 0.2690

45 57.75 57.95 27.43 31.19 30.73 26.99 0.5780 268.5 0.3170 308.7 0.2720

46 58.57 58.56 27.78 31.69 31.18 27.29 0.5841 272.0 0.3220 313.2 0.2750

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47 59.38 59.42 28.18 32.19 31.69 27.69 0.5927 276.0 0.3270 318.3 0.2790

48 60.20 60.45 28.53 32.59 32.11 28.09 0.6030 279.5 0.3310 322.5 0.2830

49 61.02 61.48 28.98 33.19 32.64 28.49 0.6133 284.0 0.3370 327.8 0.2870

50 61.83 61.90 29.34 33.59 33.10 28.89 0.6175 287.6 0.3410 332.4 0.2910

51 62.65 62.56 29.75 34.09 33.59 29.29 0.6241 291.7 0.3460 337.3 0.2950

52 63.47 63.53 30.17 34.69 34.09 29.69 0.6338 295.9 0.3520 342.3 0.2990

53 64.28 63.68 30.62 35.19 34.63 30.09 0.6353 300.4 0.3570 347.7 0.3030

54 65.10 63.90 31.04 35.69 35.13 30.49 0.6375 304.6 0.3620 352.7 0.3070

55 65.92 66.01 31.50 36.29 35.68 30.99 0.6586 309.2 0.3680 358.2 0.3120

56 66.73 66.88 31.96 36.79 36.26 31.39 0.6673 313.8 0.3730 364.0 0.3160

57 67.55 67.73 32.43 37.29 36.46 31.89 0.6758 318.5 0.3780 366.0 0.3210

58 68.37 68.71 32.92 37.79 37.43 32.39 0.6856 323.4 0.3830 375.7 0.3260

59 69.18 69.21 33.43 38.39 38.02 32.89 0.6906 328.5 0.3890 381.6 0.3310

60 70.00 69.94 33.91 38.99 38.61 33.29 0.6979 333.3 0.3950 387.5 0.3350

61 70.00 70.48 34.56 39.69 39.30 33.99 0.7033 339.8 0.4020 394.4 0.3420

62 70.00 70.19 35.24 40.19 39.86 34.39 0.7004 346.6 0.4070 400.0 0.3460

63 70.00 70.28 35.50 40.69 40.46 34.89 0.7013 349.2 0.4120 406.0 0.3510

64 70.00 70.48 36.13 41.39 41.06 35.59 0.7033 355.5 0.4190 412.0 0.3580

65 70.00 70.39 36.62 41.89 41.66 35.99 0.7024 360.4 0.4240 418.0 0.3620

66 70.00 70.29 37.15 42.39 42.16 36.49 0.7014 365.7 0.4290 423.0 0.3670

67 70.00 70.37 37.73 42.99 42.76 37.09 0.7022 371.5 0.4350 429.0 0.3730

68 70.00 70.27 38.25 43.39 43.26 37.59 0.7012 376.7 0.4390 434.0 0.3780

69 70.00 70.23 38.80 43.99 43.76 38.19 0.7008 382.2 0.4450 439.0 0.3840

70 70.00 70.31 39.34 44.49 44.26 38.69 0.7016 387.6 0.4500 444.0 0.3890

71 70.00 70.31 39.88 44.99 44.76 39.19 0.7016 393.0 0.4550 449.0 0.3940

72 70.00 70.35 43.88 48.99 48.56 45.49 0.7020 433.0 0.4950 487.0 0.4570

73 70.00 70.75 41.28 46.49 46.36 40.89 0.7060 407.0 0.4700 465.0 0.4110

74 70.00 70.35 41.38 46.49 46.36 40.99 0.7020 408.0 0.4700 465.0 0.4120

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75 70.00 70.17 41.78 46.79 46.66 41.29 0.7002 412.0 0.4730 468.0 0.4150

76 70.00 70.37 42.38 47.39 47.26 41.99 0.7022 0.418 0.4790 0.474 0.4220

77 70.00 70.39 42.88 47.69 47.56 42.39 0.7024 0.423 0.4820 0.477 0.4260

78 70.00 70.30 43.18 47.99 47.96 42.79 0.7015 0.426 0.4850 0.481 0.4300

79 70.00 70.19 43.58 48.39 48.26 43.19 0.7004 0.430 0.4890 0.484 0.4340

80 70.00 70.19 44.08 48.69 48.66 43.69 0.7004 0.435 0.4920 0.488 0.4390

81 70.00 70.23 44.48 49.19 49.16 44.39 0.7008 0.439 0.4970 0.493 0.4460

82 70.00 70.46 45.08 49.69 49.66 44.79 0.7031 0.445 0.5020 0.498 0.4500

83 70.00 70.32 45.28 49.79 49.86 44.99 0.7017 0.447 0.5030 0.500 0.4520

84 70.00 70.95 54.68 50.09 50.16 45.39 0.7080 0.541 0.5060 0.503 0.4560

85 70.00 70.24 46.08 50.39 50.46 45.79 0.7009 0.455 0.5090 0.506 0.4600

86 70.00 70.23 46.38 50.79 50.86 46.19 0.7008 0.458 0.5130 0.510 0.4640

87 70.00 70.18 46.78 51.09 51.16 46.59 0.7003 0.462 0.5160 0.513 0.4680

88 70.00 70.16 47.08 51.39 51.46 46.89 0.7001 0.465 0.5190 0.516 0.4710

89 70.00 70.18 47.48 51.69 51.76 47.29 0.7003 0.469 0.5220 0.519 0.4750

90 70.00 70.22 47.98 46.69 52.06 47.59 0.7007 0.474 0.4720 0.522 0.4780

91 70.00 70.00 48.08 52.19 52.36 47.89 0.6985 0.475 0.5270 0.525 0.4810

92 70.00 70.14 48.38 52.49 52.66 48.29 0.6999 0.478 0.5300 0.528 0.4850

93 70.00 70.18 48.68 52.69 52.96 48.59 0.7003 0.481 0.5320 0.531 0.4880

94 70.00 69.96 48.98 52.99 53.16 48.89 0.6981 0.484 0.5350 0.533 0.4910

95 70.00 69.22 49.18 53.09 53.46 49.19 0.6907 0.486 0.5360 0.536 0.4940

96 70.00 69.50 49.58 53.39 53.66 49.39 0.6935 0.490 0.5390 0.538 0.4960

97 70.00 70.36 49.78 53.59 53.96 49.69 0.7021 0.492 0.5410 0.541 0.4990

98 70.00 68.65 49.98 53.79 54.16 49.99 0.6850 0.494 0.5430 0.543 0.5020

99 70.00 70.93 50.28 54.09 54.46 50.29 0.7078 0.497 0.5460 0.546 0.5050

100 70.00 70.65 50.58 54.29 54.76 50.79 0.7050 0.500 0.5480 0.549 0.5100

101 70.00 70.34 51.08 54.69 55.16 51.09 0.7019 0.505 0.5520 0.553 0.5130

102 70.00 69.83 50.98 54.59 55.06 50.99 0.6968 0.504 0.5510 0.552 0.5120

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103 70.00 70.54 51.18 54.79 55.16 51.19 0.7039 0.506 0.5530 0.553 0.5140

104 70.00 70.31 51.48 54.99 55.46 51.49 0.7016 0.509 0.5550 0.556 0.5170

105 70.00 70.12 51.68 55.19 55.66 51.69 0.6997 0.511 0.5570 0.558 0.5190

106 70.00 70.29 52.28 55.69 56.26 52.29 0.7014 0.517 0.5620 0.564 0.5250

107 70.00 70.25 52.08 55.49 55.96 52.09 0.7010 0.515 0.5600 0.561 0.5230

108 70.00 70.25 52.28 55.69 56.16 52.29 0.7010 0.517 0.5620 0.563 0.5250

109 70.00 70.08 52.48 55.79 56.36 52.49 0.6993 0.519 0.5630 0.565 0.5270

110 70.00 70.18 52.58 55.99 56.46 52.69 0.7003 0.520 0.5650 0.566 0.5290

111 70.00 70.17 52.78 55.99 56.56 52.89 0.7002 0.522 0.5650 0.567 0.5310

112 70.00 70.35 53.08 56.29 56.86 53.19 0.7020 0.525 0.5680 0.570 0.5340

113 70.00 70.27 53.08 56.29 56.86 53.19 0.7012 0.525 0.5680 0.570 0.5340

114 70.00 70.21 53.18 56.39 56.96 53.39 0.7006 0.526 0.5690 0.571 0.5360

115 70.00 70.14 53.28 56.39 57.06 53.49 0.6999 0.527 0.5690 0.572 0.5370

116 70.00 70.38 53.48 56.69 57.26 53.69 0.7023 0.529 0.5720 0.574 0.5390

117 70.00 70.15 53.58 56.69 57.26 53.79 0.7000 0.530 0.5720 0.574 0.5400

118 70.00 70.27 53.88 56.99 57.56 54.09 0.7012 0.533 0.5750 0.577 0.5430

119 70.00 70.42 53.88 56.79 57.46 53.99 0.7027 0.533 0.5730 0.576 0.5420

120 70.00 70.38 53.88 56.99 57.56 54.09 0.7023 0.533 0.5750 0.577 0.5430

121 69.18 69.68 54.08 57.09 57.76 54.29 0.6953 0.535 0.5760 0.579 0.5450

122 68.37 68.49 54.18 57.29 57.86 54.49 0.6834 0.536 0.5780 0.580 0.5470

123 67.55 67.67 54.08 57.09 57.76 54.39 0.6752 0.535 0.5760 0.579 0.5460

124 66.73 67.25 54.28 57.29 57.86 54.59 0.6710 0.537 0.5780 0.580 0.5480

125 65.92 66.07 54.48 57.49 58.16 54.79 0.6592 0.539 0.5800 0.583 0.5500

126 65.10 64.75 52.58 57.29 57.96 54.79 0.6460 0.520 0.5780 0.581 0.5500

127 64.28 64.25 54.58 57.39 58.06 54.89 0.6410 0.540 0.5790 0.582 0.5510

128 63.47 63.91 54.48 57.29 57.86 54.79 0.6376 0.539 0.5780 0.580 0.5500

129 62.65 63.05 54.48 57.29 57.96 54.99 0.6290 0.539 0.5780 0.581 0.5520

130 61.83 61.93 54.78 57.29 57.96 54.99 0.6178 0.542 0.5780 0.581 0.5520

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302/576




159 38.15 41.09 49.68 49.39 50.46 50.49 0.4094 0.491 0.4990 0.506 0.5070

160 37.33 40.70 49.38 49.09 50.16 50.19 0.4055 0.488 0.4960 0.503 0.5040

161 36.52 40.33 48.98 48.69 49.76 49.79 0.4018 0.484 0.4920 0.499 0.5000

162 35.70 39.98 48.68 48.29 49.46 49.49 0.3983 0.481 0.4880 0.496 0.4970

163 34.88 39.64 48.38 47.89 49.06 48.79 0.3949 0.478 0.4840 0.492 0.4900

164 34.07 39.33 48.08 47.59 48.76 48.89 0.3918 0.475 0.4810 0.489 0.4910

165 33.25 38.91 47.58 47.19 48.26 48.49 0.3876 0.470 0.4770 0.484 0.4870

166 32.43 38.67 47.38 46.89 47.96 48.19 0.3852 0.468 0.4740 0.481 0.4840

167 31.62 38.37 47.08 46.49 47.66 47.89 0.3822 0.465 0.4700 0.478 0.4810

168 30.80 38.05 46.68 46.19 47.26 47.49 0.3790 0.461 0.4670 0.474 0.4770

169 29.98 37.70 46.38 45.79 46.86 47.19 0.3755 0.458 0.4630 0.470 0.4740

170 29.17 37.32 46.08 45.49 46.56 44.89 0.3717 0.455 0.4600 0.467 0.4510

171 28.35 36.94 45.68 45.09 46.16 46.59 0.3679 0.451 0.4560 0.463 0.4680

172 27.53 36.53 45.38 44.69 45.86 46.19 0.3638 0.448 0.4520 0.460 0.4640

173 26.72 36.18 45.08 44.39 45.56 45.89 0.3603 0.445 0.4490 0.457 0.4610

174 25.90 35.82 44.68 44.09 45.16 45.59 0.3567 0.441 0.4460 0.453 0.4580

175 25.08 35.48 44.38 43.79 44.86 45.19 0.3533 0.438 0.4430 0.450 0.4540

176 24.27 35.15 44.08 43.39 44.46 44.89 0.3500 0.435 0.4390 0.446 0.4510

177 23.45 34.84 43.68 43.09 44.16 44.49 0.3469 0.431 0.4360 0.443 0.4470

178 22.63 34.55 43.38 42.69 43.86 44.19 0.3440 0.428 0.4320 0.440 0.4440

179 21.82 34.24 43.08 42.39 43.46 43.89 0.3409 0.425 0.4290 0.436 0.4410

180 21.00 33.99 42.78 42.09 43.16 43.59 0.3384 0.422 0.4260 0.433 0.4380

181 21.00 33.73 42.38 41.69 42.86 43.29 0.3358 0.418 0.4220 0.430 0.4350

182 21.00 33.48 42.08 41.39 42.46 42.89 0.3333 0.415 0.4190 0.426 0.4310

183 21.00 33.24 41.78 41.09 42.16 42.59 0.3309 0.412 0.4160 0.423 0.4280

184 21.00 32.96 41.38 40.69 41.76 42.19 0.3281 0.408 0.4120 0.419 0.4240

185 21.00 32.59 40.88 40.19 41.26 41.69 0.3244 0.403 0.4070 0.414 0.4190

186 21.00 32.54 40.78 40.09 41.16 41.59 0.3239 0.402 0.4060 0.413 0.4180

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187 21.00 32.35 40.58 39.89 40.86 41.39 0.3220 0.400 0.4040 0.410 0.4160

188 21.00 32.14 40.18 39.49 40.56 40.99 0.3199 0.396 0.4000 0.407 0.4120

189 21.00 31.90 39.88 39.19 40.26 40.69 0.3175 0.393 0.3970 0.404 0.4090

190 21.00 31.75 39.58 38.89 39.96 40.39 0.3160 0.390 0.3940 0.401 0.4060

191 21.00 31.56 39.38 38.59 39.66 40.09 0.3141 0.388 0.3910 0.398 0.4030

192 21.00 31.37 39.08 38.39 39.36 39.79 0.3122 0.385 0.3890 0.395 0.4000

193 21.00 31.19 38.78 38.09 39.06 39.49 0.3104 0.382 0.3860 0.392 0.3970

194 21.00 31.01 38.48 37.79 38.76 39.19 0.3086 0.379 0.3830 0.389 0.3940

195 21.00 30.83 38.18 37.49 38.46 38.89 0.3068 0.376 0.3800 0.386 0.3910

196 21.00 30.65 37.88 37.19 38.26 38.59 0.3050 0.373 0.3770 0.384 0.3880

197 21.00 30.46 37.58 36.99 37.86 38.29 0.3031 0.370 0.3750 0.380 0.3850

198 21.00 30.29 37.38 36.69 37.66 38.09 0.3014 0.368 0.3720 0.378 0.3830

199 21.00 30.12 37.08 36.39 37.36 37.79 0.2997 0.365 0.3690 0.375 0.3800

200 21.00 29.96 36.78 36.19 37.06 37.49 0.2981 0.362 0.3670 0.372 0.3770

201 21.00 29.73 36.58 35.79 36.66 37.09 0.2958 0.360 0.3630 0.368 0.3730

202 21.00 29.65 36.38 35.69 36.56 36.99 0.2950 0.358 0.3620 0.367 0.3720

203 21.00 29.49 36.19 35.39 36.26 36.69 0.2934 0.3561 0.3590 0.364 0.3690

204 21.00 29.34 35.94 35.19 36.06 36.49 0.2919 0.3536 0.3570 0.362 0.3670

205 21.00 29.20 35.69 34.89 35.86 36.19 0.2905 0.3511 0.3540 0.360 0.3640

206 21.00 29.05 35.43 34.69 35.54 35.89 0.2890 0.3485 0.3520 0.357 0.3610

207 21.00 28.91 35.19 34.49 35.28 35.59 0.2876 346.1 0.3500 354.2 0.3580

208 21.00 28.76 34.94 34.19 35.04 35.39 0.2861 343.6 0.3470 351.8 0.3560

209 21.00 28.64 34.72 33.99 34.82 35.19 0.2849 341.4 0.3450 349.6 0.3540

210 21.00 28.50 34.48 33.79 34.59 34.99 0.2835 339.0 0.3430 347.3 0.3520

211 21.00 28.37 34.25 33.49 34.36 34.69 0.2822 336.7 0.3400 345.0 0.3490

212 21.00 28.25 34.03 33.29 34.11 34.49 0.2810 334.5 0.3380 342.5 0.3470

213 21.00 28.12 33.79 33.09 33.88 34.19 0.2797 332.1 0.3360 340.2 0.3440

214 21.00 27.99 33.58 32.89 33.68 33.99 0.2784 330.0 0.3340 338.2 0.3420

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215 21.00 27.87 33.36 32.69 33.46 33.79 0.2772 327.8 0.3320 336.0 0.3400

216 21.00 27.76 33.15 32.49 33.25 33.59 0.2761 325.7 0.3300 333.9 0.3380

217 21.00 27.64 32.93 32.29 33.03 33.39 0.2749 323.5 0.3280 331.7 0.3360

218 21.00 27.53 32.73 32.09 32.83 33.09 0.2738 321.5 0.3260 329.7 0.3330

219 21.00 27.42 32.52 31.89 32.62 32.89 0.2727 319.4 0.3240 327.6 0.3310

220 21.00 27.31 32.32 31.69 32.42 32.69 0.2716 317.4 0.3220 325.6 0.3290

221 21.00 27.21 32.13 31.49 32.23 32.49 0.2706 315.5 0.3200 323.7 0.3270

222 21.00 27.10 31.93 31.29 32.04 32.29 0.2695 313.5 0.3180 321.8 0.3250

223 21.00 27.00 31.74 31.09 31.84 32.09 0.2685 311.6 0.3160 319.8 0.3230

224 21.00 26.89 31.55 30.99 31.64 31.89 0.2674 309.7 0.3150 317.8 0.3210

225 21.00 26.80 31.37 30.79 31.46 31.69 0.2665 307.9 0.3130 316.0 0.3190

226 21.00 26.70 31.18 30.59 31.28 31.49 0.2655 306.0 0.3110 314.2 0.3170

227 21.00 26.59 30.99 30.39 31.08 31.39 0.2644 304.1 0.3090 312.2 0.3160

228 21.00 26.50 30.83 30.29 30.92 31.19 0.2635 302.5 0.3080 310.6 0.3140

229 21.00 26.41 30.65 30.09 30.75 30.99 0.2626 300.7 0.3060 308.9 0.3120

230 21.00 26.31 30.46 29.89 30.56 30.79 0.2616 298.8 0.3040 307.0 0.3100

231 21.00 26.23 30.30 29.79 30.41 30.59 0.2608 297.2 0.3030 305.5 0.3080

232 21.00 26.13 30.14 29.59 30.23 30.39 0.2598 295.6 0.3010 303.7 0.3060

233 21.00 26.05 29.98 29.39 30.07 30.29 0.2590 294.0 0.2990 302.1 0.3050

234 21.00 25.96 29.82 29.29 29.91 30.09 0.2581 292.4 0.2980 300.5 0.3030

235 21.00 25.88 29.66 29.09 29.75 29.89 0.2573 290.8 0.2960 298.9 0.3010

236 21.00 25.79 29.50 28.99 29.59 29.79 0.2564 289.2 0.2950 297.3 0.3000

237 21.00 25.69 29.30 28.79 29.39 29.59 0.2554 287.2 0.2930 295.3 0.2980

238 21.00 25.62 29.17 28.69 29.26 29.39 0.2547 285.9 0.2920 294.0 0.2960

239 21.00 25.56 29.05 28.59 29.14 29.29 0.2541 284.7 0.2910 292.8 0.2950

240 21.00 25.49 28.90 28.39 28.98 29.19 0.2534 283.2 0.2890 291.2 0.2940

241 21.82 25.42 28.76 28.29 28.83 28.99 0.2527 281.8 0.2880 289.7 0.2920

242 22.63 25.36 28.62 28.09 28.70 28.89 0.2521 280.4 0.2860 288.4 0.2910

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243 23.45 25.29 28.48 27.99 28.56 28.69 0.2514 279.0 0.2850 287.0 0.2890

244 24.27 25.23 28.35 27.89 28.43 28.59 0.2508 277.7 0.2840 285.7 0.2880

245 25.08 25.34 28.22 27.79 28.29 28.49 0.2519 276.4 0.2830 284.3 0.2870

246 25.90 26.12 28.08 27.59 28.16 28.29 0.2597 275.0 0.2810 283.0 0.2850

247 26.72 27.16 27.89 27.49 28.03 28.19 0.2701 273.1 0.2800 281.7 0.2840

248 27.53 27.48 27.83 27.49 27.94 28.09 0.2733 272.5 0.2800 280.8 0.2830

249 28.35 28.33 27.71 27.39 27.85 27.89 0.2818 271.3 0.2790 279.9 0.2810

250 29.17 28.90 27.60 27.29 27.78 27.79 0.2875 270.2 0.2780 279.2 0.2800

251 29.98 29.81 27.49 27.29 27.73 27.69 0.2966 269.1 0.2780 278.7 0.2790

252 30.80 30.94 27.39 27.29 27.69 27.59 0.3079 268.1 0.2780 278.3 0.2780

253 31.62 31.78 27.30 27.29 27.68 27.49 0.3163 267.2 0.2780 278.2 0.2770

254 32.43 32.57 27.22 27.29 27.69 27.39 0.3242 266.4 0.2780 278.3 0.2760

255 33.25 33.40 27.15 27.29 27.71 27.29 0.3325 265.7 0.2780 278.5 0.2750

256 34.07 34.29 27.09 27.39 27.76 27.29 0.3414 265.1 0.2790 279.0 0.2750

257 34.88 35.13 27.05 27.49 27.82 27.19 0.3498 264.7 0.2800 279.6 0.2740

258 35.70 35.82 27.02 27.49 27.89 27.19 0.3567 264.4 0.2800 280.3 0.2740

259 36.52 36.77 27.00 27.59 28.00 27.09 0.3662 264.2 0.2810 281.4 0.2730

260 37.33 37.53 27.00 27.79 28.10 27.09 0.3738 264.2 0.2830 282.4 0.2730

261 38.15 38.32 27.02 27.89 28.21 27.09 0.3817 264.4 0.2840 283.5 0.2730

262 38.97 38.90 27.04 27.99 28.34 27.09 0.3875 264.6 0.2850 284.8 0.2730

263 39.78 39.43 27.08 28.09 28.47 27.09 0.3928 265.0 0.2860 286.1 0.2730

264 40.60 40.77 27.14 28.29 28.61 27.19 0.4062 265.6 0.2880 287.5 0.2740

265 41.42 41.56 27.21 28.49 28.76 27.19 0.4141 266.3 0.2900 289.0 0.2740

266 42.23 42.26 27.30 28.69 28.94 27.29 0.4211 267.2 0.2920 290.8 0.2750

267 43.05 43.39 27.40 28.79 29.13 27.39 0.4324 268.2 0.2930 292.7 0.2760

268 43.87 44.08 27.52 28.99 29.33 27.49 0.4393 269.4 0.2950 294.7 0.2770

269 44.68 44.64 27.64 29.19 29.52 27.59 0.4449 270.6 0.2970 296.6 0.2780

270 45.50 45.29 27.77 29.39 29.71 27.69 0.4514 271.9 0.2990 298.5 0.2790

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271 46.32 46.50 27.92 29.69 29.91 27.79 0.4635 273.4 0.3020 300.5 0.2800

272 47.13 47.28 28.08 29.89 30.17 27.99 0.4713 275.0 0.3040 303.1 0.2820

273 47.95 47.95 28.25 30.19 30.38 28.09 0.4780 276.7 0.3070 305.2 0.2830

274 48.77 48.90 28.44 30.39 30.62 28.19 0.4875 278.6 0.3090 307.6 0.2840

275 49.58 49.49 28.63 30.69 30.90 28.39 0.4934 280.5 0.3120 310.4 0.2860

276 50.40 50.38 28.85 30.99 31.20 28.59 0.5023 282.7 0.3150 313.4 0.2880

277 51.22 51.42 29.06 31.29 31.46 28.79 0.5127 284.8 0.3180 316.0 0.2900

278 52.03 52.10 29.27 31.59 31.75 29.19 0.5195 286.9 0.3210 318.9 0.2940

279 52.85 53.03 29.51 31.89 32.05 29.19 0.5288 289.3 0.3240 321.9 0.2940

280 53.67 53.94 29.76 32.29 32.39 29.49 0.5379 291.8 0.3280 325.3 0.2970

281 54.48 54.69 30.02 32.69 32.77 29.79 0.5454 294.4 0.3320 329.1 0.3000

282 55.30 55.35 30.30 33.09 33.13 29.99 0.5520 297.2 0.3360 332.7 0.3020

283 56.12 56.08 30.60 33.49 33.54 30.29 0.5593 300.2 0.3400 336.8 0.3050

284 56.93 56.95 30.89 33.89 33.89 30.59 0.5680 303.1 0.3440 340.3 0.3080

285 57.75 57.90 31.18 34.19 34.25 30.79 0.5775 306.0 0.3470 343.9 0.3100

286 58.57 58.65 31.20 34.59 34.63 31.09 0.5850 306.2 0.3510 347.7 0.3130

287 59.38 58.25 32.01 35.19 35.21 31.49 0.5810 314.3 0.3570 353.5 0.3170

288 60.20 57.95 32.23 35.39 35.46 31.79 0.5780 316.5 0.3590 356.0 0.3200

289 61.02 59.87 32.51 35.79 35.77 32.09 0.5972 319.3 0.3630 359.1 0.3230

290 61.83 62.27 32.87 36.29 36.23 32.39 0.6212 322.9 0.3680 363.7 0.3260

291 62.65 62.79 33.22 36.69 36.68 32.79 0.6264 326.4 0.3720 368.2 0.3300

292 63.47 61.55 33.61 37.19 37.18 33.19 0.6140 330.3 0.3770 373.2 0.3340

293 64.28 62.83 33.99 37.59 37.63 33.49 0.6268 334.1 0.3810 377.7 0.3370

294 65.10 64.36 34.30 38.09 38.11 33.89 0.6421 337.2 0.3860 382.5 0.3410

295 65.92 66.14 34.79 38.59 38.64 34.29 0.6599 342.1 0.3910 387.8 0.3450

296 66.73 66.06 35.23 39.19 39.18 34.69 0.6591 346.5 0.3970 393.2 0.3490

297 67.55 67.69 35.65 39.69 39.70 35.19 0.6754 350.7 0.4020 398.4 0.3540

298 68.37 68.53 36.11 40.29 40.36 35.59 0.6838 355.3 0.4080 0.405 0.3580

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299 69.18 68.65 36.58 40.89 40.96 36.09 0.6850 360.0 0.4140 0.411 0.3630

300 70.00 69.59 37.07 41.49 41.56 36.59 0.6944 364.9 0.4200 0.417 0.3680

301 70.00 70.23 37.55 42.09 42.06 36.99 0.7008 369.7 0.4260 0.422 0.3720

302 70.00 70.01 38.07 42.69 42.66 37.49 0.6986 0.3749 0.4320 0.428 0.3770

303 70.00 70.24 38.58 43.19 43.26 37.99 0.7009 0.380 0.4370 0.434 0.3820

304 70.00 70.28 39.08 43.79 43.86 38.49 0.7013 0.385 0.4430 0.440 0.3870

305 70.00 70.25 39.58 44.29 44.36 38.99 0.7010 0.390 0.4480 0.445 0.3920

306 70.00 70.29 40.18 44.89 44.96 39.49 0.7014 0.396 0.4540 0.451 0.3970

307 70.00 70.24 40.58 45.49 45.56 40.09 0.7009 0.400 0.4600 0.457 0.4030

308 70.00 70.29 41.08 45.89 45.96 40.59 0.7014 0.405 0.4640 0.461 0.4080

309 70.00 70.20 41.58 46.49 46.56 41.09 0.7005 0.410 0.4700 0.467 0.4130

310 70.00 70.21 42.08 46.99 47.06 41.69 0.7006 0.415 0.4750 0.472 0.4190

311 70.00 70.31 42.68 47.39 47.46 42.19 0.7016 0.421 0.4790 0.476 0.4240

312 70.00 70.24 43.08 47.89 47.96 42.69 0.7009 0.425 0.4840 0.481 0.4290

313 70.00 70.14 43.68 48.39 48.46 43.19 0.6999 0.431 0.4890 0.486 0.4340

314 70.00 70.37 44.08 48.69 48.86 43.69 0.7022 0.435 0.4920 0.490 0.4390

315 70.00 70.18 44.58 49.19 49.26 44.19 0.7003 0.440 0.4970 0.494 0.4440

316 70.00 70.20 45.08 49.59 49.66 44.59 0.7005 0.445 0.5010 0.498 0.4480

317 70.00 70.18 45.48 49.99 50.06 45.09 0.7003 0.449 0.5050 0.502 0.4530

318 70.00 70.13 45.98 50.29 50.46 45.59 0.6998 0.454 0.5080 0.506 0.4580

319 70.00 70.15 46.38 50.69 50.86 45.99 0.7000 0.458 0.5120 0.510 0.4620

320 70.00 70.16 46.78 51.09 51.26 46.49 0.7001 0.462 0.5160 0.514 0.4670

331 70.00 70.14 50.58 54.29 54.66 50.39 0.6999 0.500 0.5480 0.548 0.5060

332 70.00 70.16 50.88 54.49 54.96 50.69 0.7001 0.503 0.5500 0.551 0.5090

333 70.00 70.17 51.08 54.69 55.16 50.99 0.7002 0.505 0.5520 0.553 0.5120

334 70.00 70.16 51.38 54.99 55.36 51.29 0.7001 0.508 0.5550 0.555 0.5150

335 70.00 70.21 51.58 55.19 55.66 51.59 0.7006 0.510 0.5570 0.558 0.5180

336 70.00 70.20 51.88 55.39 55.86 51.79 0.7005 0.513 0.5590 0.560 0.5200

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337 70.00 70.20 52.08 55.49 56.06 52.09 0.7005 0.515 0.5600 0.562 0.5230

338 70.00 70.13 52.28 55.69 54.96 52.29 0.6998 0.517 0.5620 0.551 0.5250

339 70.00 70.04 52.48 55.89 56.46 52.59 0.6989 0.519 0.5640 0.566 0.5280

340 70.00 70.20 52.68 56.09 56.66 52.79 0.7005 0.521 0.5660 0.568 0.5300

341 70.00 70.19 52.98 56.29 56.86 52.99 0.7004 0.524 0.5680 0.570 0.5320

342 70.00 70.13 53.08 56.39 56.96 53.19 0.6998 0.525 0.5690 0.571 0.5340

343 70.00 70.15 53.28 56.59 56.96 53.39 0.7000 0.527 0.5710 0.571 0.5360

344 70.00 70.18 53.48 56.69 56.56 53.59 0.7003 0.529 0.5720 0.567 0.5380

345 70.00 70.16 53.68 56.89 56.96 53.79 0.7001 0.531 0.5740 0.571 0.5400

346 70.00 70.14 53.78 56.99 57.66 53.99 0.6999 0.532 0.5750 0.578 0.5420

347 70.00 70.16 53.98 57.19 57.86 54.09 0.7001 0.534 0.5770 0.580 0.5430

348 70.00 70.24 54.08 57.29 57.96 54.29 0.7009 0.535 0.5780 0.581 0.5450

349 70.00 70.20 54.28 57.39 58.06 54.49 0.7005 0.537 0.5790 0.582 0.5470

350 70.00 70.21 54.48 57.49 58.26 54.59 0.7006 0.539 0.5800 0.584 0.5480

351 70.00 70.21 54.58 57.69 58.36 54.79 0.7006 0.540 0.5820 0.585 0.5500

352 70.00 70.24 54.68 57.79 58.46 54.89 0.7009 0.541 0.5830 0.586 0.5510

353 70.00 70.18 54.88 57.89 58.66 55.09 0.7003 0.543 0.5840 0.588 0.5530

354 70.00 70.16 54.98 57.99 58.76 54.19 0.7001 0.544 0.5850 0.589 0.5440

355 70.00 70.34 55.08 58.09 58.86 55.29 0.7019 0.545 0.5860 0.590 0.5550

356 70.00 70.05 55.28 58.49 58.96 55.49 0.6990 0.547 0.5900 0.591 0.5570

357 70.00 70.03 55.28 58.29 59.06 55.59 0.6988 0.547 0.5880 0.592 0.5580

358 70.00 70.19 55.38 58.39 59.16 55.69 0.7004 0.548 0.5890 0.593 0.5590

359 70.00 70.29 55.48 58.49 59.26 55.79 0.7014 0.549 0.5900 0.594 0.5600

360 70.00 69.92 55.58 58.49 59.36 55.89 0.6977 0.550 0.5900 0.595 0.5610

361 69.18 69.27 55.68 58.59 59.46 55.99 0.6912 0.551 0.5910 0.596 0.5620

362 68.37 68.50 55.78 58.69 59.46 56.09 0.6835 0.552 0.5920 0.596 0.5630

363 67.55 67.62 55.88 58.69 59.56 56.19 0.6747 0.553 0.5920 0.597 0.5640

364 66.73 66.91 55.98 58.79 59.66 56.29 0.6676 0.554 0.5930 0.598 0.5650

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365 65.92 66.12 55.98 58.79 59.66 56.39 0.6597 0.554 0.5930 0.598 0.5660

366 65.10 65.23 56.08 58.79 59.66 56.39 0.6508 0.555 0.5930 0.598 0.5660

367 64.28 64.46 56.08 58.69 59.56 56.49 0.6431 0.555 0.5920 0.597 0.5670

368 63.47 63.79 56.18 58.69 59.56 56.59 0.6364 0.556 0.5920 0.597 0.5680

369 62.65 62.85 56.18 58.59 59.46 56.59 0.6270 0.556 0.5910 0.596 0.5680

370 61.83 62.08 56.18 58.49 59.46 56.59 0.6193 0.556 0.5900 0.596 0.5680

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K.3 Random Vibration Test Results K TESTS RESULTS

K.3 Random Vibration Test Results

The following images are of the vibration prole that was applied to the satellite for the

high and low spectrums. The green line is the planned vibration spectrum and the red

line is the vibration spectrum that was actually applied to the test pod. The red line is

within the error bounds specied by the test plan.

Figure 99: High level spectrum

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K.3 Random Vibration Test Results K TESTS RESULTS

Figure 100: Low level spectrum

The following gures are resonance plots for the h

AUSAT Final Report

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Transcript of AUSAT Final Report